3. Synthesis of decentralized adaptive gain robust controllers

In this section, on the basis of the result derived in Section 2, an LMI-based design method of decentralized adaptive gain robust state feedback controllers for a class of uncertain large-scale interconnected systems is suggested. The design problem of the decentralized adaptive gain robust controller under consideration can also be reduced to the feasibility of LMIs, and the allowable perturbation region of uncertainties is also discussed.

#### 3.1. Problem statement

For Case 1 in this example, one can see from Table 1 that the adaptive gain robust controller

Case 1) <sup>4</sup>:<sup>2584</sup> � <sup>10</sup>�<sup>2</sup> 1.0160 � <sup>10</sup>�<sup>2</sup> Case 2) <sup>9</sup>:<sup>7403</sup> � <sup>10</sup>�<sup>2</sup> 1.0038 � <sup>10</sup>�<sup>1</sup>

between the time response and the desired one generated by the nominal system ("nominal" in figures) is small. But for the result of Case 2), we find that the robust controller with the parameter <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 achieves more desirable performance. Additionally, one can see from Figures 2(a) and 4(a) that by selecting the design parameter ϑ the proposed adaptive gain robust controller can adjust the magnitude of the control input. In this example, the magnitude of the control input for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup><sup>2</sup> is suppressed comparing with one for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup>

However, the online adjustment way of the design parameter ϑ for the purpose of improving transient behavior and avoiding excessive control input cannot to developed, and thus it is an

In this section, an LMI-based design scheme of the centralized adaptive gain robust state feedback controller for a class of uncertain linear systems has been proposed, and by simple numerical simulations, the effectiveness of the proposed robust control strategy has been

Therefore, the effectiveness of the proposed adaptive gain robust controller is shown.

, that is, the error

. (a) The time histories of

<sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102 <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup>

0 0.5 1 1.5 2

Time *t*

Case 1)

Case 2) Nominal

0

0.2

 0.4 0.6 0.8

Lyapunov function

(a) (b)

1

1.2

.

for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> is more desirable comparing with one for <sup>ϑ</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � 102

Figure 4. Time histories of the control inputs and the Lyapunov function for <sup>ϑ</sup> <sup>¼</sup> <sup>5</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup>

important problem of our research subjects.

Table 1. The performance index Ie.

0 0.5 1 1.5 2

Time *t*

u(t) and u tð Þ, (b) The time histories of V(x, t) and Vð Þ x; t .

Case 1) Case 2) Nominal

198 Adaptive Robust Control Systems

Control input

2.4. Summary

Consider the uncertain large-scale interconnected system composed of N subsystems described as

$$\frac{d}{dt}\mathbf{x}\_i(t) = A\_{ii}(t)\mathbf{x}\_i(t) + \sum\_{\substack{j=1\\j\neq i}}^N A\_{ij}(t)\mathbf{x}\_j(t) + B\_i u\_i(t),\tag{35}$$

where xið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup><sup>i</sup> and uið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup><sup>i</sup> (<sup>i</sup> <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, <sup>N</sup> ) are the vectors of the state and the control input for the <sup>i</sup>th subsystem, respectively, and xðÞ¼ <sup>t</sup> <sup>x</sup><sup>T</sup> <sup>1</sup> ð Þ<sup>t</sup> ; <sup>⋯</sup>; <sup>x</sup><sup>T</sup> <sup>N</sup> ð Þ<sup>t</sup> � �<sup>T</sup> is the state of the overall system. The matrices Aiið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and Aijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> in Eq. (35) are given by

$$\begin{aligned} A\_{i\bar{i}}(t) &= A\_{i\bar{i}} + \Delta\_{i\bar{i}}(t), \\ A\_{i\bar{j}}(t) &= A\_{i\bar{j}} + \Delta\_{i\bar{j}}(t). \end{aligned} \tag{36}$$

In Eqs. (35) and (36), the matrices Aii ∈ R<sup>n</sup>i�n<sup>i</sup> , Aij ∈ R<sup>n</sup>i�n<sup>j</sup> , and Bi ∈ R<sup>n</sup>i�m<sup>i</sup> denote the nominal values of the system, and matrices <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> show unknown parameters which satisfy Δ<sup>T</sup> iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ii Ini and Δ<sup>T</sup> ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ij Inj , respectively. Note that the elements of these unknown parameters are Lebesgue measurable [1, 34]. For Eq. (35), the nominal subsystem, ignoring the unknown parameters, is given by

$$\frac{d}{dt}\overline{\mathbf{x}}\_i(t) = A\_{i\overline{i}}\overline{\mathbf{x}}\_i(t) + \sum\_{\substack{j=1\\j\neq i}}^N A\_{i\overline{j}}\overline{\mathbf{x}}\_j(t) + B\_i\overline{u}\_i(t),\tag{37}$$

where <sup>x</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup><sup>i</sup> and uið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup><sup>i</sup> are the vectors of the state and the control input for the <sup>i</sup>th nominal subsystem, respectively. Furthermore, the control input for the nominal subsystem of Eq. (37) is determined as

$$
\overline{u}\_i(t) = -K\_i \overline{\mathbf{x}}\_i(t),\tag{38}
$$

d dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup>

d dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup>

Therefore, if the matrix inequality

AKi þ αiI<sup>n</sup> � �<sup>T</sup>

fixed gain Ki as Ki¼

AiiY<sup>i</sup> � BiWiBT

X N

P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>X</sup>

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>X</sup>

N

j ¼ 1 j 6¼ i

for the nominal subsystem of Eq. (37). Namely, from Eq. (46), the matrix inequality

� � <sup>þ</sup> <sup>X</sup>

is considered. In Eq. (48), α<sup>i</sup> ∈ R is a positive scalar and is selected by designers.

<sup>i</sup> <sup>þ</sup> <sup>2</sup>αiY<sup>i</sup> <sup>þ</sup> <sup>X</sup>

N

j ¼ 1 j 6¼ i

Thus, by applying Lemma 1.2 (Schur complement) to Eq. (49), we find that the matrix inequal-

holds, then the following relation for the time derivative of Vð Þ x; t is satisfied:

μi

N

j ¼ 1 j 6¼ i

PiAijA<sup>T</sup>

Now, as with Section 2, we derive a decentralized controller with pole placement constraints

N

j ¼ 1 j 6¼ i

μi PiAijA<sup>T</sup>

μi

PiAijA<sup>T</sup>

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

1 μji

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>&</sup>lt; <sup>0</sup>, <sup>∀</sup>xð Þ<sup>t</sup> 6¼ <sup>0</sup>: (47)

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

<sup>Δ</sup> P�<sup>1</sup>

<sup>μ</sup>ijAijA<sup>T</sup>

<sup>i</sup> Pi. Then for the matrix inequality of Eq. (48), by pre- and post-

ij <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

1 μji

j ¼ 1 j 6¼ i

1 μji

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

1 μji In 1

CCA

In < 0 (46)

I<sup>n</sup> þ Q<sup>i</sup> < 0, (48)

YiY<sup>i</sup> þ YiQiY<sup>i</sup> < 0: (49)

<sup>i</sup> and W<sup>i</sup> ∈ R<sup>m</sup>i�m<sup>i</sup> and define the

, it can be obtained that

xið Þt : (45)

N

X N

μi xT

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT

ijPixið Þt

http://dx.doi.org/10.5772/intechopen.71733

(44)

201

j ¼ 1 j 6¼ i

i¼1

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

<sup>þ</sup> <sup>X</sup> N

The inequality of Eq. (44) can also be rewritten as

X N

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

AT Ki

i¼1

X N

1 μij xT <sup>j</sup> ð Þt xjð Þt :

0

BB@

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>X</sup>

d

P<sup>i</sup> þ P<sup>i</sup> AKi þ αiI<sup>n</sup>

We introduce symmetric positive definite matrices Yi¼

ii � BiW<sup>T</sup>

multiplying both sides of the matrix inequality of Eq. (48) by Y<sup>i</sup>

<sup>i</sup> BT

<sup>Δ</sup>WiBT

<sup>i</sup> <sup>þ</sup> <sup>Y</sup>iA<sup>T</sup>

ity of Eq. (49) is equivalent to the following LMI:

j ¼ 1 j 6¼ i

where Ki ∈ R<sup>m</sup>i�n<sup>i</sup> is a fixed gain matrix. From Eqs. (37) and (38), the following nominal closedloop subsystem is obtained:

$$\frac{d}{dt}\overline{\mathbf{x}}\_i(t) = A\_{K\_i}\overline{\mathbf{x}}\_i(t) + \sum\_{j=1 \atop j \neq i}^{N} A\_{i\overline{j}}\overline{\mathbf{x}}\_j(t),\tag{39}$$

where AKi ¼ <sup>Δ</sup> Aii � BiKi.

Now, by using symmetric positive definite matrices P<sup>i</sup> ∈ R<sup>n</sup>i�n<sup>i</sup> , we consider the quadratic function

$$\mathcal{V}(\overline{\mathbf{x}},t) \stackrel{\Delta}{=} \sum\_{i=1}^{N} \mathcal{V}\_i(\overline{\mathbf{x}}\_i, t)\_\prime \tag{40}$$

$$\mathcal{V}\_{i}(\overline{\mathbf{x}}\_{i},t) \stackrel{\Delta}{=} \overline{\mathbf{x}}\_{i}^{T}(t)\mathcal{P}\_{i}\overline{\mathbf{x}}\_{i}(t),\tag{41}$$

as a Lyapunov function candidate. For the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t of Eq. (41), its time derivative along the trajectory of the nominal closed-loop subsystem of Eq. (39) is given by

$$\frac{d}{dt}\mathcal{V}\_i(\overline{\mathbf{x}}\_i, t) = \overline{\mathbf{x}}\_i^T(t) \Big( A\_{K\_i}^T \mathcal{P}\_i + \mathcal{P}\_i A\_{K\_i} \Big) \overline{\mathbf{x}}\_i(t) + \sum\_{j=1 \atop j \neq i}^N \mathfrak{X}\_i^T(t) \mathcal{P}\_i A\_{i\overline{j}} \overline{\mathbf{x}}\_j(t). \tag{42}$$

For the second term on the right side of Eq. (42), by using the well-known relation of Eq. (20), we can obtain the following relation:

$$\frac{d}{dt}\mathcal{V}\_{i}(\overline{\mathbf{x}}\_{i},t) \leq \overline{\mathbf{x}}\_{i}^{T}(t) \Big(A\_{K\_{i}}^{T}\mathcal{P}\_{i} + \mathcal{P}\_{i}A\_{K\_{i}}\Big)\overline{\mathbf{x}}\_{i}(t) + \sum\_{j=1 \atop j\neq i}^{N} \mu\_{ij}\overline{\mathbf{x}}\_{i}^{T}(t)\mathcal{P}\_{i}A\_{\overline{\eta}}A\_{\overline{\eta}}^{T}\mathcal{P}\overline{\mathbf{x}}\_{i}(t) + \sum\_{j=1 \atop j\neq i}^{N} \frac{1}{\mu\_{\overline{\eta}}}\overline{\mathbf{x}}\_{j}^{T}(t)\overline{\mathbf{x}}\_{\overline{\eta}}(t). \tag{43}$$

From Eqs. (40) and (43), we have

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems http://dx.doi.org/10.5772/intechopen.71733 201

$$\begin{split} \frac{d}{dt} \mathcal{V}(\overline{\mathbf{x}}, t) \leq & \sum\_{i=1}^{N} \overline{\mathbf{x}}\_{i}^{T}(t) \Big( A\_{K\_{i}}^{T} \mathcal{P}\_{i} + \mathcal{P}\_{i} A\_{K\_{i}} \Big) \overline{\mathbf{x}}\_{i}(t) + \sum\_{i=1}^{N} \sum\_{j=1}^{N} \mu\_{i} \overline{\mathbf{x}}\_{i}^{T}(t) \mathcal{P}\_{i} A\_{i\bar{j}} A\_{i\bar{j}}^{T} \mathcal{P}\_{i} \overline{\mathbf{x}}\_{i}(t) \\ &+ \sum\_{i=1}^{N} \sum\_{j=1}^{N} \frac{1}{\mu\_{i\bar{j}}} \overline{\mathbf{x}}\_{j}^{T}(t) \overline{\mathbf{x}}\_{\bar{j}}(t). \end{split} \tag{44}$$

The inequality of Eq. (44) can also be rewritten as

$$\frac{d}{dt}\mathcal{V}(\overline{\mathbf{x}},t) \leq \sum\_{i=1}^{N} \overline{\mathbf{x}}\_{i}^{T}(t) \left( A\_{\mathcal{K}\_{i}}^{T} \mathcal{P}\_{i} + \mathcal{P}\_{i} A\_{\mathcal{K}\_{i}} + \sum\_{\begin{subarray}{c} j=1\\ j\neq i \end{subarray}}^{N} \mu\_{i} \mathcal{P}\_{i} A\_{\overline{i}j} A\_{\overline{i}j}^{T} \mathcal{P}\_{i} + \sum\_{j=1}^{N} \frac{1}{\mu\_{ji}} I\_{n} \right) \overline{\mathbf{x}}\_{i}(t). \tag{45}$$

Therefore, if the matrix inequality

d

d

Now, by using symmetric positive definite matrices P<sup>i</sup> ∈ R<sup>n</sup>i�n<sup>i</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

P<sup>i</sup> þ PiAKi � �

dt <sup>x</sup>iðÞ¼ <sup>t</sup> AKi

<sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>Δ</sup> <sup>X</sup>

<sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ¼ ; <sup>t</sup> <sup>Δ</sup>

N

i¼1

xT

as a Lyapunov function candidate. For the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t of Eq. (41), its time derivative along the trajectory of the nominal closed-loop subsystem of Eq. (39) is given by

For the second term on the right side of Eq. (42), by using the well-known relation of Eq. (20),

<sup>μ</sup>ijx<sup>T</sup>

<sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijA<sup>T</sup>

j ¼ 1 j 6¼ i

2x<sup>T</sup>

ijPxiðÞþ <sup>t</sup> <sup>X</sup>

N

1 μij xT

<sup>j</sup> ð Þt xjð Þt : (43)

j ¼ 1 j 6¼ i

P<sup>i</sup> þ PiAKi � �

> <sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

> > j ¼ 1 j 6¼ i

Eq. (37) is determined as

200 Adaptive Robust Control Systems

loop subsystem is obtained:

where AKi

function

d

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>

¼

<sup>Δ</sup> Aii � BiKi.

d

we can obtain the following relation:

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

From Eqs. (40) and (43), we have

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ¼ ; <sup>t</sup> <sup>x</sup><sup>T</sup>

dt <sup>x</sup>iðÞ¼ <sup>t</sup> Aiixið Þþ <sup>t</sup> <sup>X</sup>

N

j ¼ 1 j 6¼ i

where <sup>x</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup><sup>i</sup> and uið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup><sup>i</sup> are the vectors of the state and the control input for the <sup>i</sup>th nominal subsystem, respectively. Furthermore, the control input for the nominal subsystem of

where Ki ∈ R<sup>m</sup>i�n<sup>i</sup> is a fixed gain matrix. From Eqs. (37) and (38), the following nominal closed-

<sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

Aijxjð Þþ t Biuið Þt , (37)

Aijxjð Þt , (39)

V<sup>i</sup> x<sup>i</sup> ð Þ ; t , (40)

<sup>i</sup> ð Þt Pixið Þt , (41)

, we consider the quadratic

<sup>i</sup> ð Þt PiAijxjð Þt : (42)

uiðÞ¼� t Kixið Þt , (38)

$$A\_{\mathcal{K}\_i}^T \mathcal{P}\_i + \mathcal{P}\_i A\_{\mathcal{K}\_i} + \sum\_{j=1 \atop j \neq i}^N \mu\_i \mathcal{P}\_i A\_{\vec{\eta}i} A\_{\vec{\eta}}^T \mathcal{P}\_i + \sum\_{j=1 \atop j \neq i}^N \frac{1}{\mu\_{ji}} I\_n < 0 \tag{46}$$

holds, then the following relation for the time derivative of Vð Þ x; t is satisfied:

$$\frac{d}{dt}\mathcal{V}(\overline{\mathbf{x}},t) < 0, \quad \forall \overline{\mathbf{x}}(t) \neq 0. \tag{47}$$

Now, as with Section 2, we derive a decentralized controller with pole placement constraints for the nominal subsystem of Eq. (37). Namely, from Eq. (46), the matrix inequality

$$\left(A\_{\mathbb{K}i} + a\_i I\_{\mathbb{n}}\right)^T \mathcal{P}\_i + \mathcal{P}\_i \{A\_{\mathbb{K}i} + a\_i I\_{\mathbb{n}}\} + \sum\_{\substack{j=1\\j\neq i}}^N \mu\_i \mathcal{P}\_i A\_{\vec{\mu}} A\_{\vec{\mu}}^T \mathcal{P}\_i + \sum\_{\substack{j=1\\j\neq i}}^N \frac{1}{\mu\_{ji}} I\_{\mathbb{n}} + \mathcal{Q}\_i < 0,\tag{48}$$

is considered. In Eq. (48), α<sup>i</sup> ∈ R is a positive scalar and is selected by designers.

We introduce symmetric positive definite matrices Yi¼ <sup>Δ</sup> P�<sup>1</sup> <sup>i</sup> and W<sup>i</sup> ∈ R<sup>m</sup>i�m<sup>i</sup> and define the fixed gain Ki as Ki¼ <sup>Δ</sup>WiBT <sup>i</sup> Pi. Then for the matrix inequality of Eq. (48), by pre- and postmultiplying both sides of the matrix inequality of Eq. (48) by Y<sup>i</sup> , it can be obtained that

$$A\_{\vec{u}}\mathcal{Y}\_{i} - B\_{i}\mathcal{W}\_{i}B\_{i}^{T} + \mathcal{Y}\_{i}A\_{\vec{u}}^{T} - B\_{i}\mathcal{W}\_{i}^{T}B\_{i}^{T} + 2a\_{i}\mathcal{Y}\_{i} + \sum\_{\substack{j=1\\j\neq i}}^{N} \mu\_{\vec{\eta}}A\_{\vec{\eta}}A\_{\vec{\eta}}^{T} + \sum\_{j=1}^{N} \frac{1}{\mu\_{ji}}\mathcal{Y}\_{i}\mathcal{Y}\_{i} + \mathcal{Y}\_{i}\mathcal{Q}\_{i}\mathcal{Y}\_{i} < 0. \tag{49}$$

Thus, by applying Lemma 1.2 (Schur complement) to Eq. (49), we find that the matrix inequality of Eq. (49) is equivalent to the following LMI:

$$
\begin{pmatrix}
\Lambda\_i \left( \mathcal{V}\_i, \mathcal{W}\_i, \mu\_{ij} \right) & \Theta\_i(\mathcal{V}\_i) \\
\star & -\Gamma\_i \left( \mu\_{ij} \right)
\end{pmatrix} < 0. \tag{50}
$$

Π<sup>i</sup> ξii; εij; σij � � Ξ<sup>i</sup>

if x<sup>T</sup>

if x<sup>T</sup>

, Ξ<sup>i</sup> ∈ R<sup>n</sup>i�ð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup><sup>i</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiB<sup>T</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiB<sup>T</sup>

N

εijPiAijAT

j ¼ 1 j 6¼ i

ii <sup>I</sup>n; <sup>ε</sup>1iIn; <sup>ε</sup>2iIn; <sup>⋯</sup>; <sup>ε</sup><sup>i</sup>�<sup>1</sup>iIn; <sup>ε</sup><sup>i</sup>þ<sup>1</sup>iIn; <sup>⋯</sup>; <sup>ε</sup><sup>N</sup> iIn; <sup>σ</sup>1ir<sup>⋆</sup>

In; ⋯; σ<sup>N</sup> <sup>i</sup>r<sup>⋆</sup>

In; �

N i In �

the time-varying adjustable parameters θið Þt ∈ R are determined as

ii <sup>p</sup> <sup>∥</sup>Pixið Þ<sup>t</sup> ∥∥xið Þ<sup>t</sup> <sup>∥</sup>

ii <sup>p</sup> <sup>∥</sup>Pixið Þ<sup>t</sup> ∥∥xið Þ<sup>t</sup> <sup>∥</sup> ϑix<sup>T</sup>

<sup>i</sup> ð Þt xið Þt

P<sup>i</sup> þ PiAKi

1 A,

<sup>Δ</sup> diag <sup>ξ</sup>iir<sup>⋆</sup>

where V<sup>i</sup> x<sup>i</sup> ð Þ ; t is a quadratic function given by

<sup>⋯</sup>; <sup>σ</sup><sup>i</sup>�1ir<sup>⋆</sup>

i�1i

<sup>I</sup>n; <sup>σ</sup><sup>i</sup>þ1ir<sup>⋆</sup>

iþ1i

Then, the overall close-loop system composed of N closed-loop subsystems is asymptotically stable.

<sup>V</sup>ð Þ¼ <sup>x</sup>; <sup>t</sup> <sup>Δ</sup> <sup>X</sup>

<sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ¼ ; <sup>t</sup> <sup>Δ</sup>

P<sup>i</sup> þ PiAKi

X N

j ¼ 1 j 6¼ i

Proof. In order to prove Theorem 3, the following Lyapunov function candidate is introduced by using symmetric positive definite matrices P<sup>i</sup> ∈ R<sup>n</sup>i�n<sup>i</sup> which satisfy the LMIs of (50):

N

i¼1

xT

We can obtain the following relation for the time derivative of the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t of

� �xið Þþ <sup>t</sup> 2xið Þ<sup>t</sup> <sup>P</sup>iΔiið Þ<sup>t</sup> <sup>x</sup>ið Þ<sup>t</sup>

Aij <sup>þ</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> � �xjð Þ� <sup>t</sup> <sup>2</sup>θið Þ<sup>t</sup> <sup>x</sup><sup>T</sup>

� � <sup>þ</sup> <sup>ξ</sup>iiPiP<sup>i</sup> <sup>þ</sup> <sup>X</sup>

<sup>i</sup> Pixið Þt

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiB<sup>T</sup>

ffiffiffiffiffi r⋆

ffiffiffiffiffi r⋆

<sup>Δ</sup> AT Ki

xT

8 >>>><

>>>>:

where matrices Π<sup>i</sup> ξi; εij; σij � �∈ R<sup>n</sup>i�n<sup>i</sup>

<sup>Π</sup><sup>i</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � �<sup>¼</sup>

<sup>Δ</sup> I<sup>n</sup> I<sup>n</sup> ⋯ I<sup>n</sup> zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ <sup>0</sup> <sup>2</sup><sup>N</sup> �<sup>1</sup>

Ξi¼

@

<sup>Ω</sup><sup>i</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � � <sup>¼</sup>

<sup>θ</sup>ið Þ¼<sup>t</sup> <sup>Δ</sup>

given by

Eq. (58):

d

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ¼ ; <sup>t</sup> <sup>x</sup><sup>T</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

<sup>þ</sup> 2x<sup>T</sup> <sup>i</sup> ð Þt P<sup>i</sup>

<sup>⋆</sup> �Ω<sup>i</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � � ! <sup>&</sup>lt; <sup>0</sup>, (54)

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

<sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>ix<sup>T</sup>

<sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>ix<sup>T</sup>

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

V<sup>i</sup> x<sup>i</sup> ð Þ ; t , (57)

<sup>i</sup> ð Þt Pixið Þt : (58)

<sup>i</sup> ð Þt xið Þt ,

http://dx.doi.org/10.5772/intechopen.71733

<sup>i</sup> ð Þt xið Þt ,

σijPiPi,

1i In; σ2ir<sup>⋆</sup> 2i

: (56)

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiKixið Þ<sup>t</sup> : (59)

, and Ω<sup>i</sup> ξii; εij; σij � �∈ Rð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup>i�ð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup><sup>i</sup> are

(55)

203

In Eq. (50), matrices <sup>Λ</sup><sup>i</sup> <sup>Y</sup>i; <sup>W</sup>i; <sup>μ</sup>ij � �<sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> , <sup>Θ</sup>ið Þ <sup>Y</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�<sup>N</sup> <sup>n</sup><sup>i</sup> , and <sup>Γ</sup><sup>i</sup> <sup>μ</sup>ij � �<sup>∈</sup> <sup>R</sup><sup>N</sup> <sup>n</sup>i�<sup>N</sup> <sup>n</sup><sup>i</sup> are given by

$$
\Lambda\_i \left( \mathcal{Y}\_i, \mathcal{W}\_i, \mu\_{ij} \right) \stackrel{\Delta}{=} A\_{ii} \mathcal{Y}\_i - B\_i \mathcal{W}\_i \mathcal{B}\_i^T + \mathcal{Y}\_i A\_{ii}^T - B\_i \mathcal{W}\_i^T \mathcal{B}\_i^T + 2 \alpha\_i \mathcal{Y}\_i + \sum\_{j=1 \atop j \neq i}^N \mu\_{ij} A\_{ij} A\_{ij}^T,
$$

$$
\Theta\_i(\mathcal{Y}\_i) \stackrel{\Delta}{=} \left( \overbrace{\mathcal{Y}\_i \mathcal{Y}\_i \dots \mathcal{Y}\_i}^{N} \right),
$$

$$
\Gamma\_i \left( \mu\_{ij} \right) \stackrel{\Delta}{=} \text{diag} \left( \mathcal{Q}\_i^{-1}, \mu\_{1i} I\_n, \mu\_{2i} I\_n, \dots, \mu\_{i-1} I\_n, \mu\_{i+1} I\_n, \dots, \mu\_{Ni} I\_n \right). \tag{51}
$$

Therefore, if matrices <sup>Y</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>W</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>i�m<sup>i</sup> and positive scalars <sup>μ</sup>ij exist, the nominal closed-loop subsystem is asymptotically stable, and the fixed gain matrix Ki is determined as Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup> <sup>i</sup> Y�<sup>1</sup> i .

Now, by using the fixed gain matrix Ki ∈ R<sup>m</sup>i�n<sup>i</sup> which is designed for the nominal subsystem, we define the control input

$$
\mu\_i(t) \stackrel{\Lambda}{=} -(1 + \Theta\_i(t))\mathcal{K}\_i \mathbf{x}\_i(t), \tag{52}
$$

where <sup>θ</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> is an adjustable time-varying parameter. From Eqs. (35) and (52), the uncertain closed-loop subsystem can be obtained as

$$\frac{d}{dt}\mathbf{x}\_i(t) = A\_{K\_i}\mathbf{x}\_i(t) + \Delta\_{ii}(t)\mathbf{x}\_i(t) + \sum\_{i=1}^{N} \left(A\_{\vec{\eta}} + \Delta\_{\vec{\eta}}(t)\right)\mathbf{x}\_j(t) - \theta\_i(t)\mathcal{B}\_i\mathbf{K}\_i\mathbf{x}\_i(t). \tag{53}$$

From the above discussion, the designed objective in this section is to determine the decentralized robust control of Eq. (52) such that the resultant overall system achieves robust stability. That is to design the adjustable time-varying parameter <sup>θ</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> such that asymptotical stability of the overall system composed of N subsystems of Eq. (53) is guaranteed.

#### 3.2. Decentralized variable gain controllers

The following theorem shows sufficient conditions for the existence of the proposed decentralized adaptive gain robust control system.

Theorem 3: Consider the uncertain large-scale interconnected system of Eq. (35) and the control input of Eq. (52).

For a given positive constant ϑi, if positive constants ξii, σij, and εij exist which satisfy the LMIs

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems http://dx.doi.org/10.5772/intechopen.71733 203

$$
\begin{pmatrix}
\Pi\_i(\xi\_{ii}, \varepsilon\_{\vec{\eta}}, \sigma\_{\vec{\eta}}) & \Xi\_i \\
\star & -\Omega\_i(\xi\_{ii}, \varepsilon\_{\vec{\eta}}, \sigma\_{\vec{\eta}})
\end{pmatrix} < 0,
\tag{54}
$$

the time-varying adjustable parameters θið Þt ∈ R are determined as

Λ<sup>i</sup> Yi; Wi; μij � �

∈ R<sup>n</sup>i�n<sup>i</sup>

<sup>Δ</sup> AiiY<sup>i</sup> � BiWiB<sup>T</sup>

1 A,

<sup>i</sup> ; μ1<sup>i</sup>

<sup>x</sup>ið Þþ <sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup>

overall system composed of N subsystems of Eq. (53) is guaranteed.

In; μ2<sup>i</sup>

0 B@

� �

¼

<sup>Δ</sup> Y<sup>i</sup> Y<sup>i</sup> … Y<sup>i</sup> zfflfflfflfflfflfflffl}|fflfflfflfflfflfflffl{ <sup>N</sup> 0

<sup>Δ</sup> diag Q�<sup>1</sup>

@

tain closed-loop subsystem can be obtained as

3.2. Decentralized variable gain controllers

ralized adaptive gain robust control system.

dt <sup>x</sup>iðÞ¼ <sup>t</sup> AKi

In Eq. (50), matrices Λ<sup>i</sup> Yi; Wi; μij

202 Adaptive Robust Control Systems

Θið Þ¼ Y<sup>i</sup>

Γ<sup>i</sup> μij � � ¼

Λ<sup>i</sup> Yi; Wi; μij � �

by

Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup>

of Eq. (52).

<sup>i</sup> Y�<sup>1</sup> i .

we define the control input

d

Θið Þ Y<sup>i</sup>

, <sup>Θ</sup>ið Þ <sup>Y</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�<sup>N</sup> <sup>n</sup><sup>i</sup>

� �

ii � BiW<sup>T</sup>

In; <sup>μ</sup><sup>i</sup>þ1<sup>i</sup>

<sup>i</sup> B<sup>T</sup>

� �: (51)

1

, and Γ<sup>i</sup> μij � �

In; ⋯; μ<sup>N</sup> <sup>i</sup>

uið Þ¼<sup>t</sup> <sup>Δ</sup> �ð Þ <sup>1</sup> <sup>þ</sup> <sup>θ</sup>ið Þ<sup>t</sup> Kixið Þ<sup>t</sup> , (52)

<sup>i</sup> <sup>þ</sup> <sup>2</sup>αiY<sup>i</sup> <sup>þ</sup> <sup>X</sup>

In

Aij <sup>þ</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> � �xjð Þ� <sup>t</sup> <sup>θ</sup>ið Þ<sup>t</sup> BiKixið Þ<sup>t</sup> : (53)

CA <sup>&</sup>lt; <sup>0</sup>: (50)

N

j ¼ 1 j 6¼ i

∈ R<sup>N</sup> <sup>n</sup>i�<sup>N</sup> <sup>n</sup><sup>i</sup> are given

<sup>μ</sup>ijAijA<sup>T</sup> ij ,

⋆ �Γ<sup>i</sup> μij

<sup>i</sup> <sup>þ</sup> <sup>Y</sup>iA<sup>T</sup>

In; <sup>⋯</sup>; <sup>μ</sup><sup>i</sup>�1<sup>i</sup>

Therefore, if matrices <sup>Y</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>W</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>m</sup>i�m<sup>i</sup> and positive scalars <sup>μ</sup>ij exist, the nominal closed-loop subsystem is asymptotically stable, and the fixed gain matrix Ki is determined as

Now, by using the fixed gain matrix Ki ∈ R<sup>m</sup>i�n<sup>i</sup> which is designed for the nominal subsystem,

where <sup>θ</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> is an adjustable time-varying parameter. From Eqs. (35) and (52), the uncer-

N

i¼1

From the above discussion, the designed objective in this section is to determine the decentralized robust control of Eq. (52) such that the resultant overall system achieves robust stability. That is to design the adjustable time-varying parameter <sup>θ</sup>ið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> such that asymptotical stability of the

The following theorem shows sufficient conditions for the existence of the proposed decent-

Theorem 3: Consider the uncertain large-scale interconnected system of Eq. (35) and the control input

For a given positive constant ϑi, if positive constants ξii, σij, and εij exist which satisfy the LMIs

$$\boldsymbol{\Theta}\_{i}(t) \stackrel{\Delta}{=} \begin{cases} \frac{\sqrt{\rho^{\star}\_{ii}} \| \mathcal{P}\_{i} \mathbf{x}\_{i}(t) \| \| \mathbf{x}\_{i}(t) \|}{\mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{B}\_{i} \mathcal{W}\_{i} \mathbf{B}\_{i}^{T} \mathcal{P}\_{i} \mathbf{x}\_{i}(t)} & \text{if} \quad \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{B}\_{i} \mathcal{W}\_{i} \mathbf{B}\_{i}^{T} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) \geq \boldsymbol{\mathcal{S}}\_{i} \mathbf{x}\_{i}^{T}(t) \mathbf{x}\_{i}(t),\\ \frac{\sqrt{\rho^{\star}\_{ii}} \| \mathbf{P}\_{i} \mathbf{x}\_{i}(t) \| \| \mathbf{x}\_{i}(t) \|}{\mathbf{x}\_{i} \mathbf{x}\_{i}^{T}(t) \mathbf{x}\_{i}(t)} & \text{if} \quad \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{B}\_{i} \mathcal{W}\_{i} \mathbf{B}\_{i}^{T} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) < \boldsymbol{\mathcal{S}}\_{i} \mathbf{x}\_{i}^{T}(t) \mathbf{x}\_{i}(t), \end{cases} \tag{55}$$

where matrices Π<sup>i</sup> ξi; εij; σij � �∈ R<sup>n</sup>i�n<sup>i</sup> , Ξ<sup>i</sup> ∈ R<sup>n</sup>i�ð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup><sup>i</sup> , and Ω<sup>i</sup> ξii; εij; σij � �∈ Rð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup>i�ð Þ <sup>2</sup><sup>N</sup> �<sup>1</sup> <sup>n</sup><sup>i</sup> are given by

<sup>Π</sup><sup>i</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � �<sup>¼</sup> <sup>Δ</sup> AT Ki P<sup>i</sup> þ PiAKi � � <sup>þ</sup> <sup>ξ</sup>iiPiP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N j ¼ 1 j 6¼ i εijPiAijAT ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N j ¼ 1 j 6¼ i σijPiPi, Ξi¼ <sup>Δ</sup> I<sup>n</sup> I<sup>n</sup> ⋯ I<sup>n</sup> zfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflffl{ <sup>0</sup> <sup>2</sup><sup>N</sup> �<sup>1</sup> @ 1 A, <sup>Ω</sup><sup>i</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � � <sup>¼</sup> <sup>Δ</sup> diag <sup>ξ</sup>iir<sup>⋆</sup> ii <sup>I</sup>n; <sup>ε</sup>1iIn; <sup>ε</sup>2iIn; <sup>⋯</sup>; <sup>ε</sup><sup>i</sup>�<sup>1</sup>iIn; <sup>ε</sup><sup>i</sup>þ<sup>1</sup>iIn; <sup>⋯</sup>; <sup>ε</sup><sup>N</sup> iIn; <sup>σ</sup>1ir<sup>⋆</sup> 1i In; σ2ir<sup>⋆</sup> 2i In; � <sup>⋯</sup>; <sup>σ</sup><sup>i</sup>�1ir<sup>⋆</sup> i�1i <sup>I</sup>n; <sup>σ</sup><sup>i</sup>þ1ir<sup>⋆</sup> iþ1i In; ⋯; σ<sup>N</sup> <sup>i</sup>r<sup>⋆</sup> N i In � : (56)

Then, the overall close-loop system composed of N closed-loop subsystems is asymptotically stable.

Proof. In order to prove Theorem 3, the following Lyapunov function candidate is introduced by using symmetric positive definite matrices P<sup>i</sup> ∈ R<sup>n</sup>i�n<sup>i</sup> which satisfy the LMIs of (50):

$$\mathcal{V}(\mathbf{x},t) \stackrel{\Delta}{=} \sum\_{i=1}^{N} \mathcal{V}\_i(\mathbf{x}\_i, t)\_\prime \tag{57}$$

where V<sup>i</sup> x<sup>i</sup> ð Þ ; t is a quadratic function given by

$$\mathcal{V}\_{i}(\mathbf{x}\_{i},t) \stackrel{\Delta}{=} \mathbf{x}\_{i}^{T}(t)\mathcal{P}\_{i}\mathbf{x}\_{i}(t). \tag{58}$$

We can obtain the following relation for the time derivative of the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t of Eq. (58):

$$\begin{split} \frac{d}{dt} \mathcal{V}\_{i}(\mathbf{x}\_{i},t) &= \mathbf{x}\_{i}^{T}(t) \Big( \mathbf{A}\_{K\_{i}}^{T} \mathcal{P}\_{i} + \mathcal{P}\_{i} \mathbf{A}\_{K\_{i}} \Big) \mathbf{x}\_{i}(t) + 2 \mathbf{x}\_{i}(t) \mathcal{P}\_{i} \Delta\_{\mathrm{ii}}(t) \mathbf{x}\_{i}(t) \\ &+ 2 \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \sum\_{\begin{subarray}{c} j=1\\ j \neq i \end{subarray}}^{N} \Big( \mathbf{A}\_{\overline{\mathbf{y}}} + \Delta\_{\overline{\mathbf{y}}}(t) \Big) \mathbf{x}\_{i}(t) - 2 \boldsymbol{\Theta}\_{i}(t) \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{B}\_{i} \mathbf{K}\_{i} \mathbf{x}\_{i}(t). \end{split} \tag{59}$$

Firstly, we consider the case of x<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT <sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>ix<sup>T</sup> <sup>i</sup> ð Þt xið Þt . In this case, one can see from the relations Δ<sup>T</sup> iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ii Ini and Δ<sup>T</sup> ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ij Inj , the well-known inequality of Eq. (20), and Lemma 1.1 that the following relation for the quadratic function V<sup>i</sup> xi ð Þ ; t of Eq. (58) can be obtained:

d

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>X</sup>

N

0

BBB@

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>X</sup>

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>X</sup>

N

j ¼ 1 j 6¼ i

ii Ini and <sup>Δ</sup><sup>T</sup>

time-varying parameter θið Þt of Eq. (55) into Eq. (59), we have

<sup>ε</sup>ijxið Þ<sup>t</sup> <sup>P</sup>iAijAT

P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>ξ</sup>iix<sup>T</sup>

<sup>j</sup> ð Þt xjð Þ� t 2

P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>ξ</sup>iix<sup>T</sup>

Therefore, we see from Eqs. (57) and (67) that the following inequality:

<sup>ε</sup>ijxið Þ<sup>t</sup> <sup>P</sup>iAijA<sup>T</sup>

N

εijPiAijAT

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

> 1 εji

<sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>ix<sup>T</sup>

ij Inj

N

1 εij xT

ii <sup>p</sup> <sup>∥</sup>Pixið Þ<sup>t</sup> ∥∥xið Þ<sup>t</sup> <sup>∥</sup> ϑix<sup>T</sup>

!

<sup>i</sup> ð Þt xið Þt

<sup>i</sup> P<sup>i</sup> and θið Þt is a positive scalar function. Therefore, we find that the following

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>r</sup><sup>⋆</sup>

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jð Þþ <sup>t</sup> <sup>X</sup>

j ¼ 1 j 6¼ i

ijPixið Þþ <sup>t</sup> <sup>X</sup>

ffiffiffiffiffi r⋆

The last term on the right side of Eq. (66) is less than 0 because the matrix Ki ∈ R<sup>m</sup>i�n<sup>i</sup> is defined

N

1 εij xT

j ¼ 1 j 6¼ i

ijPixið Þþ <sup>t</sup> <sup>X</sup>

j ¼ 1 j 6¼ i

1 εji I<sup>n</sup> þ X N

<sup>I</sup><sup>n</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>r</sup><sup>⋆</sup>

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jð Þþ <sup>t</sup> <sup>X</sup>

j ¼ 1 j 6¼ i

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

<sup>σ</sup>ijPiP<sup>i</sup> <sup>þ</sup> <sup>X</sup>

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>&</sup>lt; <sup>0</sup>, <sup>∀</sup>xð Þ<sup>t</sup> 6¼ <sup>0</sup>: (65)

<sup>σ</sup>ijPiP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

N

r⋆ ij σji

<sup>i</sup> ð Þt xið Þt . In this case, by using the

<sup>i</sup> ð Þt PiPixið Þt

, and Eq. (20) and substituting the adjustable

σijx<sup>T</sup>

<sup>i</sup> ð Þt PiBiKixið Þt :

ii ξii xT <sup>i</sup> ð Þt xið Þt

N

j ¼ 1 j 6¼ i

ii ξii xT <sup>i</sup> ð Þt xið Þt

N

j ¼ 1 j 6¼ i

σijx<sup>T</sup>

<sup>i</sup> ð Þt PiPixið Þt

xT

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

http://dx.doi.org/10.5772/intechopen.71733

r⋆ ji σji In 1

CCCA<sup>x</sup>ið Þ<sup>t</sup> :

I<sup>n</sup> < 0 (64)

(63)

205

(66)

(67)

j ¼ 1 j 6¼ i

εijPiAijA<sup>T</sup>

d

holds, then the following relation for the time derivative of Vð Þ x; t is satisfied:

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT

ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> AT Ki

AT Ki

Therefore, if the matrix inequality

Next, we consider the case of x<sup>T</sup>

iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

þ<sup>X</sup> N

þ<sup>X</sup> N

> j ¼ 1 j 6¼ i

r⋆ ij σij xT

relation for the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t is satisfied:

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

þ<sup>X</sup> N

þ<sup>X</sup> N

> j ¼ 1 j 6¼ i

r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þt :

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

relations Δ<sup>T</sup>

d

as Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup>

d

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>

$$\begin{split} \frac{d}{dt} \mathcal{V}\_{i}(\mathbf{x}\_{i},t) \leq & \mathbf{x}\_{i}^{T}(t) \Big( \mathbf{A}\_{ki}^{T}\mathcal{P}\_{i} + \mathcal{P}\_{i}\mathbf{A}\_{K\_{i}} \Big) \mathbf{x}\_{i}(t) + 2\sqrt{\rho\_{ii}^{\star\star}} \| \mathcal{P}\_{i}\mathbf{x}\_{i}(t) \| \| \mathbf{x}\_{i}(t) \| \\ & + \sum\_{\begin{subarray}{c} j=1\\ j\neq i \end{subarray}}^{N} \varepsilon\_{ij} \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{A}\_{ij} \mathbf{A}\_{ij}^{T} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) + \sum\_{j=1}^{N} \frac{1}{\varepsilon\_{ij}} \mathbf{x}\_{i}^{T}(t) \mathbf{x}\_{j}(t) + \sum\_{j=1}^{N} \sigma\_{ij} \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{x}\_{i}(t) \\ & + \sum\_{\begin{subarray}{c} j=1\\ j\neq i \end{subarray}}^{N} \frac{\rho\_{ij}^{\star}}{\sigma\_{ij}} \mathbf{x}\_{j}^{T}(t) \mathbf{x}\_{j}(t) - 2\theta\_{i}(t) \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathbf{B}\_{i} \mathbf{K}\_{i} \mathbf{x}\_{i}(t). \end{split} \tag{60}$$

Substituting the adjustable time-varying parameter θið Þt of Eq. (55) into Eq. (60) gives

d dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> AT Ki P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>2</sup> ffiffiffiffiffi r⋆ ii <sup>q</sup> k k Pixið Þt k k xið Þt þ X N j ¼ 1 j 6¼ i εijx<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT ijPixið Þþ <sup>t</sup> <sup>X</sup> N j ¼ 1 j 6¼ i 1 εij xT <sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jð Þþ <sup>t</sup> <sup>X</sup> N j ¼ 1 j 6¼ i σijx<sup>T</sup> <sup>i</sup> ð Þt PiPixið Þt þ X N j ¼ 1 j 6¼ i r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þ� t 2 ffiffiffiffiffi r⋆ ii <sup>p</sup> <sup>∥</sup>Pixið Þ<sup>t</sup> ∥∥xið Þ<sup>t</sup> <sup>∥</sup> xT <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT <sup>i</sup> Pixið Þt !xT <sup>i</sup> ð Þt PiBiKixið Þt <sup>¼</sup> <sup>x</sup><sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>X</sup> N j ¼ 1 j 6¼ i εijx<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT ijPixið Þþ <sup>t</sup> <sup>X</sup> N j ¼ 1 j 6¼ i 1 εij xT <sup>j</sup> ð Þt xjð Þt þ X N j ¼ 1 j 6¼ i σijx<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup> N j ¼ 1 j 6¼ i r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þt , (61)

and, thus, we have the following inequality for the function Vð Þ x; t of Eq. (57):

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) \leq \sum\_{i=1}^{N} \mathbf{x}\_i^T(t) \left( A\_{K\_i}^T \mathcal{P}\_i + \mathcal{P}\_i A\_{K\_i} \right) \mathbf{x}\_i(t) + \sum\_{i=1}^{N} \sum\_{j=1}^{N} \varepsilon\_{ij} \mathbf{x}\_i(t) \mathcal{P}\_i A\_{ij} A\_{ij}^T \mathcal{P}\_i \mathbf{x}\_i(t)$$

$$+ \sum\_{i=1}^{N} \sum\_{j=1 \atop j \neq i}^{N} \frac{1}{\varepsilon\_{ij}} \mathbf{x}\_j^T(t) \mathbf{x}\_j(t) + \sum\_{i=1}^{N} \sum\_{j=1 \atop j \neq i}^{N} \sigma\_{ij} \mathbf{x}\_i^T(t) \mathcal{P}\_i \mathbf{x}\_i(t) + \sum\_{i=1}^{N} \sum\_{j=1 \atop j \neq i}^{N} \frac{\rho\_{ij}^{\star}}{\sigma\_{ij}} \mathbf{x}\_j^T(t) \mathbf{x}\_j(t) . \tag{62}$$

Furthermore, the inequality of Eq. (62) can be rewritten as

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems http://dx.doi.org/10.5772/intechopen.71733 205

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) \leq \sum\_{i=1}^{N} \mathbf{x}\_{i}^{T}(t) \left( A\_{K\_{i}}^{T}\mathcal{P}\_{i} + \mathcal{P}\_{i}\mathcal{A}\_{K\_{i}} + \sum\_{j=1 \atop j \neq i}^{N} \mathcal{C}\_{\overline{j}}\mathcal{P}\_{i}\mathcal{A}\_{\overline{j}i}A\_{\overline{i}\overline{j}}^{T}\mathcal{P}\_{i} + \sum\_{j=1 \atop j \neq i}^{N} \frac{1}{\mathcal{E}\_{\overline{j}i}}I\_{\mathbf{n}} \right.\\ \left. + \sum\_{j=1 \atop j \neq i}^{N} \mathcal{C}\_{\overline{j}i}\mathcal{P}\_{i} + \sum\_{j=1 \atop j \neq i}^{N} \frac{\rho\_{\overline{j}i}^{\star}}{\mathcal{C}\_{\overline{j}i}}I\_{\mathbf{n}} \right) \mathbf{x}\_{i}(t). \tag{63}$$

Therefore, if the matrix inequality

Firstly, we consider the case of x<sup>T</sup>

iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

εijx<sup>T</sup>

r⋆ ij σij xT

þ<sup>X</sup> N

þ<sup>X</sup> N

<sup>i</sup> ð Þ<sup>t</sup> AT Ki

εijx<sup>T</sup>

r⋆ ij σij xT

þ X N

þ X N

<sup>¼</sup> <sup>x</sup><sup>T</sup>

þ X N

d dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup>

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

<sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

σijx<sup>T</sup>

X N

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

þ X N

i¼1

X N

1 εij xT

Furthermore, the inequality of Eq. (62) can be rewritten as

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

j ¼ 1 j 6¼ i

the relations Δ<sup>T</sup>

204 Adaptive Robust Control Systems

obtained:

d

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>

d

dt <sup>V</sup><sup>i</sup> <sup>x</sup><sup>i</sup> ð Þ ; <sup>t</sup> <sup>≤</sup> <sup>x</sup><sup>T</sup>

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT

ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup>

xiðÞþ t 2

ijPixiðÞþ <sup>t</sup> <sup>X</sup>

Substituting the adjustable time-varying parameter θið Þt of Eq. (55) into Eq. (60) gives

xið Þþ t 2

ijPixið Þþ <sup>t</sup> <sup>X</sup>

ffiffiffiffiffi r⋆ ii

<sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

N

j ¼ 1 j 6¼ i

and, thus, we have the following inequality for the function Vð Þ x; t of Eq. (57):

P<sup>i</sup> þ PiAKi � �

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jðÞþ <sup>t</sup> <sup>X</sup>

N

X N

j ¼ 1 j 6¼ i

i¼1

xT

ii Ini and Δ<sup>T</sup>

P<sup>i</sup> þ PiAKi � �

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT

P<sup>i</sup> þ PiAKi � �

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT

<sup>j</sup> ð Þt xjð Þ� t 2

P<sup>i</sup> þ PiAKi � �

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup>

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jðÞ� <sup>t</sup> <sup>2</sup>θið Þ<sup>t</sup> <sup>x</sup><sup>T</sup>

<sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>≥</sup> <sup>ϑ</sup>ix<sup>T</sup>

ffiffiffiffiffi r⋆ ii q

> 1 εij xT

<sup>i</sup> ð Þt PiBiKixið Þt :

k k Pixið Þt k k xið Þt

<sup>i</sup> Pixið Þt

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iAijAT

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jð Þþ <sup>t</sup> <sup>X</sup>

N

σijx<sup>T</sup>

<sup>i</sup> ð Þt PiBiKixið Þt

N

1 εij xT <sup>j</sup> ð Þt xjð Þt

ijPixið Þt

r⋆ ij σij xT

<sup>j</sup> ð Þt xjð Þt : (62)

j ¼ 1 j 6¼ i

ijPixið Þþ <sup>t</sup> <sup>X</sup>

<sup>ε</sup>ijxið Þ<sup>t</sup> <sup>P</sup>iAijA<sup>T</sup>

N

X N

j ¼ 1 j 6¼ i

i¼1

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup>

<sup>i</sup> ð Þt PiPixið Þt

j ¼ 1 j 6¼ i

xT

N

j ¼ 1 j 6¼ i

ffiffiffiffiffi r⋆ ii q

> 1 εij xT

<sup>p</sup> <sup>∥</sup>Pixið Þ<sup>t</sup> ∥∥xið Þ<sup>t</sup> <sup>∥</sup>

εijx<sup>T</sup>

<sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

i¼1

σijx<sup>T</sup>

X N

j ¼ 1 j 6¼ i

!

N

j ¼ 1 j 6¼ i

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT

j ¼ 1 j 6¼ i

r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þt ,

k k Pixið Þt k k xið Þt

<sup>j</sup> ð Þ<sup>t</sup> <sup>x</sup>jð Þþ <sup>t</sup> <sup>X</sup>

N

j ¼ 1 j 6¼ i

σijx<sup>T</sup>

<sup>i</sup> ð Þt PiPixið Þt

(60)

(61)

ij Inj

Lemma 1.1 that the following relation for the quadratic function V<sup>i</sup> xi ð Þ ; t of Eq. (58) can be

<sup>i</sup> ð Þt xið Þt . In this case, one can see from

, the well-known inequality of Eq. (20), and

$$A\_{K\_i}^T \mathcal{P}\_i + \mathcal{P}\_i A\_{K\_i} + \sum\_{j=1 \atop j \neq i}^N \varepsilon\_{i\overline{j}} \mathcal{P}\_i A\_{i\overline{j}} A\_{i\overline{j}}^T \mathcal{P}\_i + \sum\_{j=1 \atop j \neq i}^N \frac{1}{\varepsilon\_{j\overline{i}}} I\_{\text{n}} + \sum\_{j=1 \atop j \neq i}^N \sigma\_{i\overline{j}} \mathcal{P}\_i \mathcal{P}\_i + \sum\_{j=1 \atop j \neq i}^N \frac{\rho\_{i\overline{j}}^{\star}}{\sigma\_{j\overline{i}}} I\_{\text{n}} < 0 \tag{64}$$

holds, then the following relation for the time derivative of Vð Þ x; t is satisfied:

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) < 0, \quad \forall \mathbf{x}(t) \neq 0. \tag{65}$$

Next, we consider the case of x<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iBiWiBT <sup>i</sup> <sup>P</sup>ixið Þ<sup>t</sup> <sup>&</sup>lt; <sup>ϑ</sup>ix<sup>T</sup> <sup>i</sup> ð Þt xið Þt . In this case, by using the relations Δ<sup>T</sup> iið Þ<sup>t</sup> <sup>Δ</sup>iið Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ii Ini and <sup>Δ</sup><sup>T</sup> ijð Þ<sup>t</sup> <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>≤</sup> <sup>r</sup><sup>⋆</sup> ij Inj , and Eq. (20) and substituting the adjustable time-varying parameter θið Þt of Eq. (55) into Eq. (59), we have

$$\begin{split} \frac{d}{dt}\mathcal{V}\_{i}(\mathbf{x}\_{i},t) \leq \mathbf{x}\_{i}^{T}(t) \Big( \mathbf{A}\_{\mathcal{K}\_{i}}^{T}\mathcal{P}\_{i} + \mathcal{P}\_{i}\mathcal{A}\_{\mathcal{K}\_{i}} \Big) \mathbf{x}\_{i}(t) + \boldsymbol{\xi}\_{i}\mathbf{x}\_{i}^{T}(t)\mathcal{P}\_{i}\mathcal{P}\_{i}\mathbf{x}\_{i}(t) + \frac{\rho\_{ii}^{\star}}{\mathcal{E}\_{ii}}\mathbf{x}\_{i}^{T}(t)\mathbf{x}\_{i}(t) \\ + \sum\_{\begin{subarray}{c}i=1\\j\neq i\end{subarray}}^{N} \varepsilon\_{i\rangle}\mathbf{x}\_{i}(t)\mathcal{P}\_{i}\mathcal{A}\_{\dot{\mathcal{W}}}\mathbf{x}\_{i\dot{\mathcal{W}}}^{T}\mathcal{P}\_{i}\mathbf{x}\_{i}(t) + \sum\_{\begin{subarray}{c}i=1\\j\neq i\end{subarray}}^{N} \mathbf{x}\_{i\dot{j}}^{T}(t)\mathbf{x}\_{i}(t) + \sum\_{\begin{subarray}{c}i=1\\j\neq i\end{subarray}}^{N} \sigma\_{i\dot{j}}\mathbf{x}\_{i}^{T}(t)\mathcal{P}\_{i}\mathcal{P}\_{i}\mathbf{x}\_{i}(t) \\ + \sum\_{\begin{subarray}{c}i=1\\j\neq i\end{subarray}}^{N} \sigma\_{i\dot{j}}^{\star}\mathbf{x}\_{i}(t)\mathbf{x}\_{i}(t) - 2\Big(\frac{\sqrt{\mu\_{ii}^{\star}}\|\mathcal{P}\_{i}\mathbf{x}\_{i}(t)\|\|\|\mathbf{x}\_{i}(t)\|\| }\right) \mathbf{x}\_{i}^{T}(t)\mathcal{P}\_{i}\mathcal{B}\_{i}\mathbf{x}\_{i}(t). \end{split} \tag{66}$$

The last term on the right side of Eq. (66) is less than 0 because the matrix Ki ∈ R<sup>m</sup>i�n<sup>i</sup> is defined as Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup> <sup>i</sup> P<sup>i</sup> and θið Þt is a positive scalar function. Therefore, we find that the following relation for the quadratic function V<sup>i</sup> x<sup>i</sup> ð Þ ; t is satisfied:

$$\begin{split} \frac{d}{dt} \mathcal{V}\_{i}(\mathbf{x}\_{i},t) \leq & \mathbf{x}\_{i}^{T}(t) \Big( A\_{K\_{i}}^{T} \mathcal{P}\_{i} + \mathcal{P}\_{i} A\_{K\_{i}} \Big) \mathbf{x}\_{i}(t) + \xi\_{i} \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) + \frac{\rho\_{i}^{\star}}{\xi\_{ii}} \mathbf{x}\_{i}^{T}(t) \mathbf{x}\_{i}(t) \\ & + \sum\_{j=1}^{N} \varepsilon\_{ij} \mathbf{x}\_{i}(t) \mathcal{P}\_{i} A\_{\bar{\eta}} A\_{\bar{\eta}}^{T} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) + \sum\_{j=1}^{N} \frac{1}{\varepsilon\_{\bar{\eta}}} \mathbf{x}\_{j}^{T}(t) \mathbf{x}\_{j}(t) + \sum\_{j=1 \atop j \neq i}^{N} \sigma\_{i\bar{\eta}} \mathbf{x}\_{i}^{T}(t) \mathcal{P}\_{i} \mathcal{P}\_{i} \mathbf{x}\_{i}(t) \\ & + \sum\_{j=1}^{N} \theta\_{\bar{\eta}}^{\star} \mathbf{x}\_{j}^{T}(t) \mathbf{x}\_{j}(t). \end{split} \tag{67}$$

Therefore, we see from Eqs. (57) and (67) that the following inequality:

d dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>X</sup> N i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> AT Ki P<sup>i</sup> þ PiAKi � �xið Þþ <sup>t</sup> <sup>X</sup> N i¼1 ξiix<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup> N i¼1 r⋆ ii ξii xT <sup>i</sup> ð Þt xið Þt þ X N i¼1 X N j ¼ 1 j 6¼ i <sup>ε</sup>ijxið Þ<sup>t</sup> PiAijA<sup>T</sup> ijPixið Þþ <sup>t</sup> <sup>X</sup> N i¼1 X N j ¼ 1 j 6¼ i 1 εij xT <sup>j</sup> ð Þt xjð Þt þ X N i¼1 X N j ¼ 1 j 6¼ i σijx<sup>T</sup> <sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup> N i¼1 X N j ¼ 1 j 6¼ i r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þt (68)

r⋆ ii ¼ ξii, r⋆

<sup>0</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � � <sup>Ξ</sup><sup>i</sup>

⋆ �Ω<sup>i</sup>

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>ξ</sup>iiPiP<sup>i</sup> <sup>þ</sup> <sup>N</sup> In <sup>þ</sup> <sup>X</sup>

the maximization of r<sup>⋆</sup>

Πi

Ξi 0¼

Ωi

should be solved.

of Eq. (52).

@

<sup>0</sup> <sup>ξ</sup>ii; <sup>ε</sup>ij; <sup>σ</sup>ij � �<sup>¼</sup>

<sup>Δ</sup> I<sup>n</sup> I<sup>n</sup> ⋯ I<sup>n</sup> zfflfflfflfflfflfflfflfflfflfflffl}|fflfflfflfflfflfflfflfflfflfflffl{ <sup>0</sup> <sup>N</sup> �<sup>1</sup>

Γ ¼

and consider the following additional condition:

constraint of Eq. (75) can be transformed into

constrained convex optimization problem

As a result, the following theorem can be obtained:

Minimize <sup>ξ</sup>ii><sup>0</sup>, <sup>ε</sup>ij><sup>0</sup>, <sup>σ</sup>ij><sup>0</sup>

ii and r<sup>⋆</sup>

<sup>Δ</sup> A<sup>T</sup> Ki Πi

1 A,

<sup>0</sup> <sup>ε</sup>ij � � <sup>¼</sup><sup>Δ</sup> diag <sup>ε</sup>1iIn; <sup>ε</sup>2iIn; <sup>⋯</sup>; <sup>ε</sup><sup>i</sup>�<sup>1</sup>iIn; <sup>ε</sup><sup>i</sup>þ<sup>1</sup>iIn; <sup>⋯</sup>; <sup>ε</sup><sup>N</sup> <sup>i</sup> ð Þ <sup>I</sup><sup>n</sup> :

Furthermore, we introduce a positive scalar <sup>λ</sup> and a complementary matrix <sup>Γ</sup><sup>∈</sup> <sup>R</sup><sup>N</sup> <sup>2</sup>

<sup>Γ</sup> � <sup>1</sup>

Γ I<sup>N</sup> <sup>2</sup> ⋆ λI<sup>N</sup> <sup>2</sup> !

Namely, we can replace the maximization problem of ξii and σij with the minimization problem of λ. From Eq. (75) and Lemma 1.2 (Schur complement), one can easily see that the

Thus, in order to design the proposed decentralized adaptive gain robust controller, the

Theorem 4: Consider the uncertain large-scale interconnected system of Eq. (35) and the control input

If positive constants ξii, εij, σij, and λ exist which satisfy the constrained convex optimization problem of Eq. (77), the adjustable time-varying parameter θið Þt is designed as Eq. (55). Then, the overall

From the relations of Eq. (71), one can find that the maximization of ξii and σij is equivalent to

ij . Then, the LMIs of Eq. (54) can be rewritten as

0

<sup>0</sup> <sup>ε</sup>ij � � ! <sup>&</sup>lt; <sup>0</sup>, (72)

εijPiAijA<sup>T</sup>

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

<sup>λ</sup> <sup>I</sup><sup>N</sup> <sup>2</sup> <sup>&</sup>gt; <sup>0</sup>: (75)

> 0: (76)

½ � λ subject to 72 ð Þ and 76 ð Þ (77)

j ¼ 1 j 6¼ i

http://dx.doi.org/10.5772/intechopen.71733

σijPiPi,

�<sup>N</sup> <sup>2</sup>

(73)

207

defined as

N

j ¼ 1 j 6¼ i

<sup>Δ</sup> diag <sup>ξ</sup>11; <sup>ξ</sup>22; <sup>⋯</sup>; <sup>ξ</sup>NN ; <sup>σ</sup>12; <sup>σ</sup>13; <sup>⋯</sup>; <sup>σ</sup>1<sup>N</sup> ; <sup>σ</sup>21; <sup>σ</sup>23; <sup>⋯</sup>; <sup>σ</sup>NN �<sup>1</sup> ð Þ, (74)

ij <sup>¼</sup> <sup>σ</sup>ij: (71)

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

can be derived. Moreover, one can easily see that the inequality of Eq. (68) can be rewritten as

$$\frac{d}{dt}\mathcal{V}(\mathbf{x},t) \leq \sum\_{i=1}^{N} \mathbf{x}\_{i}^{T}(t) \left( A\_{K\_{i}}^{T}\mathcal{P}\_{i} + \mathcal{P}\_{i}A\_{K\_{i}} + \xi\_{i\bar{t}}\mathcal{P}\_{i}\mathcal{P}\_{i} + \frac{\rho\_{i\bar{t}}^{\star}}{\xi\_{i\bar{t}}}I\_{n} + \sum\_{j=1 \atop j\neq i}^{N} \varepsilon\_{ij}\mathcal{P}\_{i}A\_{\bar{j}i}A\_{\bar{j}j}^{T}\mathcal{P}\_{i} \right)$$
 
$$+ \sum\_{\begin{subarray}{c}j=1\\j\neq i\end{subarray}}^{N} \frac{1}{\varepsilon\_{ji}}I\_{\mathbf{n}} + \sum\_{j=1 \atop j\neq i}^{N} \mathcal{P}\_{i}\mathcal{P}\_{i}\mathcal{P}\_{i} + \sum\_{j=1}^{N} \frac{\rho\_{ji}^{\star}}{\sigma\_{ji}}I\_{\mathbf{n}} \right) \mathbf{x}\_{i}(t). \tag{69}$$

Therefore, if the matrix inequality

$$\begin{split} \boldsymbol{A}\_{K}^{T}\boldsymbol{\mathcal{P}}\_{i} &+ \boldsymbol{\mathcal{P}}\_{i}\boldsymbol{A}\_{K\_{i}} + \boldsymbol{\xi}\_{i\bar{\boldsymbol{\mu}}}\boldsymbol{\mathcal{P}}\_{i}\boldsymbol{\mathcal{P}}\_{i} + \frac{\boldsymbol{\rho}\_{\bar{\boldsymbol{\mu}}}^{\star}}{\boldsymbol{\xi}\_{\bar{\boldsymbol{\mu}}}}\boldsymbol{I}\_{\text{n}} + \sum\_{j=1 \atop j\neq i}^{N} \boldsymbol{\varepsilon}\_{i\bar{\boldsymbol{\mu}}}\boldsymbol{\mathcal{P}}\_{i}\boldsymbol{A}\_{i\bar{\boldsymbol{\mu}}}\boldsymbol{A}\_{\bar{\boldsymbol{\mu}}}^{T}\boldsymbol{\mathcal{P}}\_{i} + \sum\_{j=1 \atop j\neq i}^{N} \boldsymbol{\mathcal{I}}\_{i\bar{\boldsymbol{\mu}}} \boldsymbol{\mathcal{P}}\_{i} \boldsymbol{\mathcal{P}}\_{i} \\ &+ \sum\_{j=1 \atop j\neq i}^{N} \frac{\boldsymbol{\rho}\_{\bar{\boldsymbol{\mu}}}^{\star}}{\boldsymbol{\sigma}\_{\bar{\boldsymbol{\mu}}}}\boldsymbol{I}\_{\text{n}} < 0 \end{split} \tag{70}$$

holds, then the relation of Eq. (65) for the time derivative of the function Vð Þ x; t of Eq. (57) is satisfied. Due to the 3rd and 4th terms on the left side of Eq. (70) which are positive definite, if the inequality of Eq. (70) is satisfied, then the inequality of Eq. (64) is also constantly satisfied.

For the matrix inequality of Eq. (70), by applying Lemma 1.2 (Schur complement), one can find that the matrix inequalities of Eq. (70) are equivalent to the LMIs of Eq. (54). Therefore, by solving the LMIs of Eq. (54), the adjustable time-varying parameter is given by Eq. (55), and proposed control input of Eq. (52) stabilizes the overall system of Eq. (35). Thus, the proof of Theorem 3 is completed.

Next, as mentioned in Section 2, we discuss the allowable region of the unknown parameters <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> . Thus, the following additional constraints are introduced:

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems http://dx.doi.org/10.5772/intechopen.71733 207

$$
\begin{aligned}
\rho\_{\vec{u}}^{\star} &= \xi\_{\vec{u}\nu} \\
\rho\_{\vec{\eta}}^{\star} &= \sigma\_{\vec{\eta}}.
\end{aligned}
\tag{71}
$$

From the relations of Eq. (71), one can find that the maximization of ξii and σij is equivalent to the maximization of r<sup>⋆</sup> ii and r<sup>⋆</sup> ij . Then, the LMIs of Eq. (54) can be rewritten as

$$
\begin{pmatrix}
\Pi\_{\varGamma}(\xi\_{i\bar{i}}, \varepsilon\_{\bar{i}\bar{j}}, \sigma\_{\bar{i}}) & \Sigma\_{\varGamma} \\
\star & -\varDelta\_{\varGamma}(\varepsilon\_{\bar{i}\bar{j}})
\end{pmatrix} < 0,\tag{72}
$$

$$
\begin{split}
\Pi\_{\varGamma}(\xi\_{i\bar{i}}, \varepsilon\_{\bar{i}\bar{j}}, \sigma\_{\bar{i}\bar{j}}) & \triangleq A\_{K\_{i}}^{\operatorname{T}} \mathscr{P}\_{i} + \mathscr{P}\_{i} A\_{K\_{i}} + \mathscr{E}\_{i\bar{i}} \mathscr{P}\_{i} \mathscr{P}\_{i} + \mathscr{M}\_{\operatorname{n}} + \sum\_{\substack{j=1 \\ j\neq i}}^{N} \varepsilon\_{\bar{i}\bar{j}} \mathscr{P}\_{i} A\_{\bar{i}\bar{j}} A\_{\bar{i}\bar{j}}^{\operatorname{T}} \mathscr{P}\_{i} + \sum\_{\substack{j=1 \\ j\neq i}}^{N} \sigma\_{\bar{i}\bar{j}} \mathscr{P}\_{i} \mathscr{P}\_{i} \\
\Xi\_{\varGamma}(\varepsilon\_{\bar{i}}) \overset{\operatorname{\bfd}}{=} \big( \begin{split} & \Sigma\_{\operatorname{\bfd}} \end{split}\Big|\, \begin{split} \operatorname{\bfd} & \Sigma\_{\operatorname{\bfd}} \\ & \sum\_{j=1}^{N} \end{split}
$$

$$
\begin{split} & \Sigma\_{\varGamma} \triangleq \Big( \begin{split} & \Sigma\_{\operatorname{\bfd}} \\ & \sum\_{i} \end{split} \Big|\, \begin{split} & \Sigma\_{\operatorname{\bfd}} \\ & \sum\_{i} \end{split}$$

Furthermore, we introduce a positive scalar <sup>λ</sup> and a complementary matrix <sup>Γ</sup><sup>∈</sup> <sup>R</sup><sup>N</sup> <sup>2</sup> �<sup>N</sup> <sup>2</sup> defined as

$$I \stackrel{\Delta}{=} \text{diag}(\xi\_{11}, \xi\_{22}, \dots, \xi\_{NN}, \sigma\_{12}, \sigma\_{13}, \dots, \sigma\_{1N}, \sigma\_{21}, \sigma\_{23}, \dots, \sigma\_{NN-1})\_{\prime} \tag{74}$$

and consider the following additional condition:

$$
\Gamma - \frac{1}{\lambda} I\_{\mathcal{N}^2} > 0. \tag{75}
$$

Namely, we can replace the maximization problem of ξii and σij with the minimization problem of λ. From Eq. (75) and Lemma 1.2 (Schur complement), one can easily see that the constraint of Eq. (75) can be transformed into

$$
\begin{pmatrix}
\Gamma & I\_{\mathcal{N}^2} \\
\star & \lambda I\_{\mathcal{N}^2}
\end{pmatrix} > 0. \tag{76}
$$

Thus, in order to design the proposed decentralized adaptive gain robust controller, the constrained convex optimization problem

$$\underset{\substack{\varepsilon\_{i\uparrow} > 0, \ \varepsilon\_{i\uparrow} > 0, \ \varepsilon\_{i\uparrow} > 0}}{\text{Minimize}} \quad [\lambda] \text{ subject to (72) and (76)}\tag{77}$$

should be solved.

d

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>X</sup>

206 Adaptive Robust Control Systems

N

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> AT Ki

þ X N

þ X N

d

i¼1

i¼1

dt <sup>V</sup>ð Þ <sup>x</sup>; <sup>t</sup> <sup>≤</sup> <sup>X</sup>

Therefore, if the matrix inequality

þ<sup>X</sup> N

> j ¼ 1 j 6¼ i

AT Ki

Theorem 3 is completed.

<sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup>

X N

j ¼ 1 j 6¼ i

X N

σijx<sup>T</sup>

j ¼ 1 j 6¼ i

N

i¼1 xT <sup>i</sup> ð Þ<sup>t</sup> <sup>A</sup><sup>T</sup> Ki

þ<sup>X</sup> N

> j ¼ 1 j 6¼ i

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>ξ</sup>iiPiP<sup>i</sup> <sup>þ</sup> <sup>r</sup><sup>⋆</sup>

I<sup>n</sup> < 0

r⋆ ji σji

1 εji <sup>I</sup><sup>n</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

ii ξii <sup>I</sup><sup>n</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

holds, then the relation of Eq. (65) for the time derivative of the function Vð Þ x; t of Eq. (57) is satisfied. Due to the 3rd and 4th terms on the left side of Eq. (70) which are positive definite, if the inequality of Eq. (70) is satisfied, then the inequality of Eq. (64) is also constantly satisfied.

For the matrix inequality of Eq. (70), by applying Lemma 1.2 (Schur complement), one can find that the matrix inequalities of Eq. (70) are equivalent to the LMIs of Eq. (54). Therefore, by solving the LMIs of Eq. (54), the adjustable time-varying parameter is given by Eq. (55), and proposed control input of Eq. (52) stabilizes the overall system of Eq. (35). Thus, the proof of

Next, as mentioned in Section 2, we discuss the allowable region of the unknown parameters

P<sup>i</sup> þ PiAKi � �

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup>

<sup>ε</sup>ijxið Þ<sup>t</sup> PiAijA<sup>T</sup>

0

BBB@

<sup>x</sup>ið Þþ <sup>t</sup> <sup>X</sup> N

ijPixið Þþ <sup>t</sup> <sup>X</sup>

N

i¼1

can be derived. Moreover, one can easily see that the inequality of Eq. (68) can be rewritten as

<sup>σ</sup>ijPiP<sup>i</sup> <sup>þ</sup> <sup>X</sup>

i¼1

N

i¼1

X N

j ¼ 1 j 6¼ i

<sup>P</sup><sup>i</sup> <sup>þ</sup> <sup>P</sup>iAKi <sup>þ</sup> <sup>ξ</sup>iiPiP<sup>i</sup> <sup>þ</sup> <sup>r</sup><sup>⋆</sup>

N

r⋆ ji σji In

j ¼ 1 j 6¼ i

εijPiAijAT

ii ξii

1

CCA

ijP<sup>i</sup> <sup>þ</sup> <sup>X</sup> N

. Thus, the following additional constraints are introduced:

j ¼ 1 j 6¼ i

1 εji <sup>I</sup><sup>n</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

σijPiP<sup>i</sup>

(70)

<sup>I</sup><sup>n</sup> <sup>þ</sup> <sup>X</sup> N

> j ¼ 1 j 6¼ i

εijPiAijA<sup>T</sup>

xið Þt : (69)

ijP<sup>i</sup>

ξiix<sup>T</sup>

X N

1 εij xT <sup>j</sup> ð Þt xjð Þt

j ¼ 1 j 6¼ i

r⋆ ij σij xT <sup>j</sup> ð Þt xjð Þt

<sup>i</sup> ð Þ<sup>t</sup> <sup>P</sup>iPixið Þþ <sup>t</sup> <sup>X</sup>

N

r⋆ ii ξii xT <sup>i</sup> ð Þt xið Þt

(68)

i¼1

As a result, the following theorem can be obtained:

Theorem 4: Consider the uncertain large-scale interconnected system of Eq. (35) and the control input of Eq. (52).

If positive constants ξii, εij, σij, and λ exist which satisfy the constrained convex optimization problem of Eq. (77), the adjustable time-varying parameter θið Þt is designed as Eq. (55). Then, the overall uncertain closed-loop system of Eq. (53) is asymptotically stable. Furthermore, by using the optimal solution ξ<sup>⋆</sup> ii and σ<sup>⋆</sup> ij for Eq. (77), the upper bound of unknown parameters <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> is given by

$$\begin{aligned} \rho\_{\vec{u}}^{\star} &= \xi\_{\vec{u}}^{\star}, \\ \rho\_{\vec{v}}^{\star} &= \sigma\_{\vec{v}}^{\star}. \end{aligned} \tag{78}$$

Thus, the symmetric positive definite matrices <sup>P</sup><sup>i</sup> <sup>¼</sup> <sup>Y</sup>�<sup>1</sup>

<sup>⋆</sup> <sup>1</sup>:7996 !, K<sup>1</sup> <sup>¼</sup> ð Þ <sup>8</sup>:1918 7:<sup>0723</sup> ,

<sup>⋆</sup> <sup>4</sup>:3025 !, K<sup>2</sup> <sup>¼</sup> <sup>1</sup>:<sup>1174</sup> � <sup>10</sup><sup>1</sup> �3:<sup>5221</sup> � �,

<sup>⋆</sup> <sup>3</sup>:0823 !, K<sup>3</sup> <sup>¼</sup> <sup>8</sup>:<sup>0240</sup> �8:<sup>0297</sup> � <sup>10</sup>�<sup>1</sup> � �:

, ε<sup>31</sup> ¼ 9:9709, ε<sup>32</sup> ¼ 1:9446,

Next, by solving the constrained convex optimization problem of Eq. (77), the following

, <sup>ξ</sup><sup>22</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

, <sup>ε</sup><sup>13</sup> <sup>¼</sup> <sup>5</sup>:<sup>9622</sup> � <sup>10</sup>�<sup>1</sup>

, <sup>σ</sup><sup>13</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

, <sup>σ</sup><sup>31</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

<sup>22</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

<sup>13</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

<sup>31</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

ij � � cos ð Þ <sup>π</sup> � <sup>t</sup> sin 3ð Þ :<sup>0</sup> � <sup>π</sup> � <sup>t</sup> ⋆ sin ð Þ π � t

Moreover, the design parameters ϑið Þ i ¼ 1; 2; 3 , the initial value of the uncertain large-scale system with system parameters of Eq. (79), and one of the nominal systems are selected as

uið Þt denote the lth element (l ¼ 1, 2) of the state xið Þt and the control input uið Þt for ith subsystem and one of the states xið Þt and the control input uið Þt for ith nominal subsystem.

!

Therefore, the allowable upper bound of unknown parameters is given as

, r<sup>⋆</sup>

, r<sup>⋆</sup>

, r<sup>⋆</sup>

In this example, unknown parameters <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> are chosen as

<sup>ϑ</sup><sup>1</sup> <sup>¼</sup> <sup>ϑ</sup><sup>2</sup> <sup>¼</sup> <sup>ϑ</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> and x 0ð Þ¼ x 0ð Þ¼ <sup>1</sup>:<sup>5</sup> �1:<sup>0</sup> �1:0 5:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> <sup>2</sup>:<sup>0</sup> �1:<sup>0</sup> � �<sup>T</sup>

The result of this example is shown in Figures 5 and 6. In these figures, xð Þ<sup>l</sup>

<sup>i</sup> can be calculated as

7:6854 2:0845

<sup>P</sup><sup>2</sup> <sup>¼</sup> <sup>1</sup>:<sup>0923</sup> � <sup>10</sup><sup>1</sup> �5:<sup>8891</sup>

<sup>P</sup><sup>3</sup> <sup>¼</sup> <sup>2</sup>:<sup>4946</sup> �2:<sup>4964</sup> � <sup>10</sup>�<sup>1</sup>

<sup>ξ</sup><sup>11</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

<sup>ε</sup><sup>12</sup> <sup>¼</sup> <sup>8</sup>:<sup>5122</sup> � <sup>10</sup>�<sup>1</sup>

<sup>ε</sup><sup>23</sup> <sup>¼</sup> <sup>3</sup>:<sup>1440</sup> � <sup>10</sup>�<sup>1</sup>

<sup>σ</sup><sup>12</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

<sup>σ</sup><sup>23</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

<sup>11</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

<sup>12</sup> <sup>¼</sup> <sup>3</sup>:<sup>4167</sup> � <sup>10</sup>�<sup>2</sup>

<sup>23</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

<sup>Δ</sup>iiðÞ¼ <sup>t</sup> <sup>r</sup><sup>⋆</sup>

<sup>Δ</sup>ijðÞ¼ <sup>t</sup> <sup>r</sup><sup>⋆</sup>

ii �

λ ¼ 1:0001:

r⋆

r⋆

r⋆

Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup>

<sup>i</sup> Y�<sup>1</sup>

P<sup>1</sup> ¼

solution can be obtained:

<sup>i</sup> and the fixed gain matrices

http://dx.doi.org/10.5772/intechopen.71733

,

,

,

,

,

:

.

ð Þl <sup>i</sup> ð Þt , and

<sup>i</sup> ð Þt , uið Þt , x

, <sup>ξ</sup><sup>33</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

, <sup>σ</sup><sup>21</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

, <sup>σ</sup><sup>32</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

<sup>33</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

<sup>21</sup> <sup>¼</sup> <sup>3</sup>:<sup>5524</sup> � <sup>10</sup>�<sup>2</sup>

<sup>32</sup> <sup>¼</sup> <sup>1</sup>:<sup>5590</sup> � <sup>10</sup>�<sup>1</sup>

:

,

, ε<sup>21</sup> ¼ 1:4174,

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

, r<sup>⋆</sup>

, r<sup>⋆</sup>

, r<sup>⋆</sup>

sin 5ð Þ� :0 � π � t cos 2ð Þ :0 � π � t ⋆ cos 5ð Þ :0 � π � t

!

(81)

209

(82)

(83)

(84)

#### 3.3. Illustrative examples

To demonstrate the efficiency of the proposed decentralized robust controller, an illustrative example is provided. In this example, we consider the uncertain large-scale interconnected system consisting of three two-dimensional subsystems, that is, N ¼ 3. The system parameters are given as follows:

$$\begin{aligned} A\_{11} &= \begin{pmatrix} -1.0 & 1.0 \\ 0.0 & 1.0 \end{pmatrix}, A\_{22} = \begin{pmatrix} 0.0 & 1.0 \\ -1.0 & -1.0 \end{pmatrix}, A\_{33} = \begin{pmatrix} 1.0 & 0.0 \\ 1.0 & -3.0 \end{pmatrix}, \\ B\_{1} &= \begin{pmatrix} 0.0 \\ 1.0 \end{pmatrix}, B\_{2} = \begin{pmatrix} 1.0 \\ 1.0 \end{pmatrix}, B\_{3} = \begin{pmatrix} 1.0 \\ 0.0 \end{pmatrix}, \\ A\_{12} &= \begin{pmatrix} 0.5 & 0.0 \\ 0.0 & 1.0 \end{pmatrix}, A\_{13} = \begin{pmatrix} 0.0 & 0.5 \\ 0.0 & 0.0 \end{pmatrix}, A\_{21} = \begin{pmatrix} 0.0 & 0.0 \\ 0.0 & 0.5 \end{pmatrix}, \\ A\_{23} &= \begin{pmatrix} 0.0 & 0.5 \\ 1.0 & 0.0 \end{pmatrix}, A\_{31} = \begin{pmatrix} 0.5 & 0.0 \\ 0.0 & 0.0 \end{pmatrix}, A\_{32} = \begin{pmatrix} 0.0 & 0.5 \\ 0.0 & 0.5 \end{pmatrix}. \end{aligned} \tag{79}$$

Firstly, by selecting the design parameters <sup>α</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> and <sup>Q</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> ð Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup> as α<sup>1</sup> ¼ α<sup>2</sup> ¼ α<sup>3</sup> ¼ 1:0 and Q<sup>1</sup> ¼ Q<sup>2</sup> ¼ Q<sup>3</sup> ¼ 2:0 � I<sup>2</sup> and solving LMIs of Eq. (50), we have the symmetric positive definite matrices Y<sup>i</sup> ∈ R<sup>2</sup>�<sup>2</sup> and W<sup>i</sup> ∈ R<sup>1</sup>�<sup>1</sup> , and positive scalars μij can be obtained:

$$\begin{aligned} \mathcal{Y}\_1 &= \begin{pmatrix} 1.8972 & -2.1976 \\ & \star & 8.1021 \end{pmatrix} \times 10^{-1}, & \mathcal{W}\_1 &= 3.9298, \\ \mathcal{Y}\_2 &= \begin{pmatrix} 3.4941 & 4.7825 \\ & \star & 8.8702 \end{pmatrix} \times 10^{-1}, & \mathcal{W}\_2 &= 2.2200, \\ \mathcal{Y}\_3 &= \begin{pmatrix} 4.0414 \times 10^{-1} & 3.2732 \times 10^{-2} \\ & \star & 3.2709 \times 10^{-1} \end{pmatrix}, & \mathcal{W}\_3 &= 3.2166, \\ \mu\_{12} &= 7.0526 \times 10^{-1}, & \mu\_{13} &= 4.5522 \times 10^{-1}, \ \mu\_{21} = 1.3986, \\ \mu\_{23} &= 3.2285 \times 10^{-1}, & \mu\_{31} &= 3.4477, \ \mu\_{32} = 2.0763. \end{aligned}$$

Thus, the symmetric positive definite matrices <sup>P</sup><sup>i</sup> <sup>¼</sup> <sup>Y</sup>�<sup>1</sup> <sup>i</sup> and the fixed gain matrices Ki <sup>¼</sup> <sup>W</sup>iB<sup>T</sup> <sup>i</sup> Y�<sup>1</sup> <sup>i</sup> can be calculated as

uncertain closed-loop system of Eq. (53) is asymptotically stable. Furthermore, by using the optimal

To demonstrate the efficiency of the proposed decentralized robust controller, an illustrative example is provided. In this example, we consider the uncertain large-scale interconnected system consisting of three two-dimensional subsystems, that is, N ¼ 3. The system parameters

0:0 1:0

, A<sup>33</sup> ¼

, A<sup>21</sup> <sup>¼</sup> <sup>0</sup>:0 0:<sup>0</sup>

, A<sup>32</sup> <sup>¼</sup> <sup>0</sup>:0 0:<sup>5</sup>

, W<sup>1</sup> ¼ 3:9298,

, W<sup>2</sup> ¼ 2:2200,

A, W<sup>3</sup> ¼ 3:2166,

, μ<sup>21</sup> ¼ 1:3986,

1

0:0 0:5

!

0:0 0:5

!

!

�1:0 �1:0

 ! ,

0:0

, B<sup>3</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup>

0:0 0:0

!

0:0 0:0

Firstly, by selecting the design parameters <sup>α</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>1</sup> and <sup>Q</sup><sup>i</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> ð Þ <sup>i</sup> <sup>¼</sup> <sup>1</sup>; <sup>2</sup>; <sup>3</sup> as α<sup>1</sup> ¼ α<sup>2</sup> ¼ α<sup>3</sup> ¼ 1:0 and Q<sup>1</sup> ¼ Q<sup>2</sup> ¼ Q<sup>3</sup> ¼ 2:0 � I<sup>2</sup> and solving LMIs of Eq. (50), we have the

<sup>A</sup> � <sup>10</sup>�<sup>1</sup>

<sup>A</sup> � <sup>10</sup>�<sup>1</sup>

, <sup>μ</sup><sup>13</sup> <sup>¼</sup> <sup>4</sup>:<sup>5522</sup> � <sup>10</sup>�<sup>1</sup>

, μ<sup>31</sup> ¼ 3:4477, μ<sup>32</sup> ¼ 2:0763:

1

1

<sup>4</sup>:<sup>0414</sup> � <sup>10</sup>�<sup>1</sup> <sup>3</sup>:<sup>2732</sup> � <sup>10</sup>�<sup>2</sup>

<sup>⋆</sup> <sup>3</sup>:<sup>2709</sup> � <sup>10</sup>�<sup>1</sup>

!

, A<sup>13</sup> <sup>¼</sup> <sup>0</sup>:0 0:<sup>5</sup>

, A<sup>31</sup> <sup>¼</sup> <sup>0</sup>:5 0:<sup>0</sup>

r⋆ ii <sup>¼</sup> <sup>ξ</sup><sup>⋆</sup> ii ,

r⋆ ij <sup>¼</sup> <sup>σ</sup><sup>⋆</sup>

ij for Eq. (77), the upper bound of unknown parameters <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>i</sup> and

ij : (78)

1:0 0:0

,

(79)

(80)

!

1:0 �3:0

,

:

, and positive scalars μij can be

solution ξ<sup>⋆</sup>

ii and σ<sup>⋆</sup>

<sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>n</sup>i�n<sup>j</sup> is given by

208 Adaptive Robust Control Systems

3.3. Illustrative examples

are given as follows:

obtained:

A<sup>11</sup> ¼

<sup>B</sup><sup>1</sup> <sup>¼</sup> <sup>0</sup>:<sup>0</sup>

1:0

<sup>A</sup><sup>12</sup> <sup>¼</sup> <sup>0</sup>:5 0:<sup>0</sup>

<sup>A</sup><sup>23</sup> <sup>¼</sup> <sup>0</sup>:0 0:<sup>5</sup>

Y<sup>1</sup> ¼

Y<sup>2</sup> ¼

Y<sup>3</sup> ¼

0 @

0 @

0 @

<sup>μ</sup><sup>12</sup> <sup>¼</sup> <sup>7</sup>:<sup>0526</sup> � <sup>10</sup>�<sup>1</sup>

<sup>μ</sup><sup>23</sup> <sup>¼</sup> <sup>3</sup>:<sup>2285</sup> � <sup>10</sup>�<sup>1</sup>

0:0 1:0

!

1:0 0:0

symmetric positive definite matrices Y<sup>i</sup> ∈ R<sup>2</sup>�<sup>2</sup> and W<sup>i</sup> ∈ R<sup>1</sup>�<sup>1</sup>

1:8972 �2:1976

3:4941 4:7825

⋆ 8:8702

⋆ 8:1021

!

!

�1:0 1:0 0:0 1:0

, B<sup>2</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup>

, A<sup>22</sup> ¼

1:0

!

!

$$\begin{aligned} \mathcal{P}\_1 &= \begin{pmatrix} 7.6854 & 2.0845 \\ \star & 1.7996 \end{pmatrix}, & K\_1 &= \begin{pmatrix} 8.1918 & 7.0723 \end{pmatrix}, \\ \mathcal{P}\_2 &= \begin{pmatrix} 1.0923 \times 10^1 & -5.8891 \\ \star & 4.3025 \end{pmatrix}, & K\_2 &= \begin{pmatrix} 1.1174 \times 10^1 & -3.5221 \end{pmatrix}, & \text{(81)}, \\ \mathcal{P}\_3 &= \begin{pmatrix} 2.4946 & -2.4964 \times 10^{-1} \\ \star & 3.0823 \end{pmatrix}, & K\_3 &= \begin{pmatrix} 8.0240 & -8.0297 \times 10^{-1} \end{pmatrix}. \end{aligned}$$

Next, by solving the constrained convex optimization problem of Eq. (77), the following solution can be obtained:

$$\begin{aligned} \xi\_{11} &= 3.4167 \times 10^{-2}, & \xi\_{22} &= 3.5524 \times 10^{-2}, & \xi\_{33} &= 1.5590 \times 10^{-1}, \\ \varepsilon\_{12} &= 8.5122 \times 10^{-1}, & \varepsilon\_{13} &= 5.9622 \times 10^{-1}, & \varepsilon\_{21} &= 1.4174, \\ \varepsilon\_{23} &= 3.1440 \times 10^{-1}, & \varepsilon\_{31} &= 9.9709, & \varepsilon\_{32} &= 1.9446, \\ \sigma\_{12} &= 3.4167 \times 10^{-2}, & \sigma\_{13} &= 3.4167 \times 10^{-2}, & \sigma\_{21} &= 3.5524 \times 10^{-2}, \\ \sigma\_{23} &= 3.5524 \times 10^{-2}, & \sigma\_{31} &= 1.5590 \times 10^{-1}, & \sigma\_{32} &= 1.5590 \times 10^{-1}, \\ \lambda &= 1.0001. & & & \\ \end{aligned} \tag{82}$$

Therefore, the allowable upper bound of unknown parameters is given as

$$\begin{aligned} \rho\_{11}^{\star} &= 3.4167 \times 10^{-2}, & \rho\_{22}^{\star} &= 3.5524 \times 10^{-2}, & \rho\_{33}^{\star} &= 1.5590 \times 10^{-1}, \\ \rho\_{12}^{\star} &= 3.4167 \times 10^{-2}, & \rho\_{13}^{\star} &= 3.4167 \times 10^{-2}, & \rho\_{21}^{\star} &= 3.5524 \times 10^{-2}, \\ \rho\_{23}^{\star} &= 3.5524 \times 10^{-2}, & \rho\_{31}^{\star} &= 1.5590 \times 10^{-1}, & \rho\_{32}^{\star} &= 1.5590 \times 10^{-1}. \end{aligned} \tag{83}$$

In this example, unknown parameters <sup>Δ</sup>iið Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> and <sup>Δ</sup>ijð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>2</sup>�<sup>2</sup> are chosen as

$$\begin{split} \Delta\_{ii}(t) &= \rho\_{ii}^{\star} \times \begin{pmatrix} \sin\left(5.0 \times \pi \times t\right) & -\cos\left(2.0 \times \pi \times t\right) \\ \star & \cos\left(5.0 \times \pi \times t\right) \end{pmatrix}, \\ \Delta\_{ij}(t) &= \rho\_{ij}^{\star} \times \begin{pmatrix} -\cos\left(\pi \times t\right) & \sin\left(3.0 \times \pi \times t\right) \\ \star & \sin\left(\pi \times t\right) \end{pmatrix}. \end{split} \tag{84}$$

Moreover, the design parameters ϑið Þ i ¼ 1; 2; 3 , the initial value of the uncertain large-scale system with system parameters of Eq. (79), and one of the nominal systems are selected as <sup>ϑ</sup><sup>1</sup> <sup>¼</sup> <sup>ϑ</sup><sup>2</sup> <sup>¼</sup> <sup>ϑ</sup><sup>3</sup> <sup>¼</sup> <sup>1</sup>:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> and x 0ð Þ¼ x 0ð Þ¼ <sup>1</sup>:<sup>5</sup> �1:<sup>0</sup> �1:0 5:<sup>0</sup> � <sup>10</sup>�<sup>1</sup> <sup>2</sup>:<sup>0</sup> �1:<sup>0</sup> � �<sup>T</sup> .

The result of this example is shown in Figures 5 and 6. In these figures, xð Þ<sup>l</sup> <sup>i</sup> ð Þt , uið Þt , x ð Þl <sup>i</sup> ð Þt , and uið Þt denote the lth element (l ¼ 1, 2) of the state xið Þt and the control input uið Þt for ith subsystem and one of the states xið Þt and the control input uið Þt for ith nominal subsystem.

Furthermore, the effectiveness of the proposed controller has been shown via an illustrative example. The proposed adaptive gain robust controller can be easily designed by solving a constrained convex optimization problem and adjust the magnitude of the control input for each subsystem. Therefore, we find that the proposed decentralized robust controller design

Adaptive Gain Robust Control Strategies for Uncertain Dynamical Systems

http://dx.doi.org/10.5772/intechopen.71733

211

Future research subjects include analysis of conservatism for the proposed controller design approach and extension of the proposed adaptive gain robust control strategies to uncertain systems with time delay, decentralized output/observer-based control systems,

In this chapter, firstly the centralized adaptive gain robust controller for a class of uncertain linear systems has been proposed, and through a simple numerical example, we have shown the effectiveness/usefulness for the proposed adaptive gain robust control strategy. Next, for a class of uncertain large-scale interconnected systems, we have presented an LMI-based design method of decentralized adaptive gain robust controllers. In the proposed controller robust synthesis, advantages are as follows: the proposed adaptive gain robust controller can achieve satisfactory transient behavior and/or avoid the excessive control input, that is, the proposed robust controller with adjustable time-varying parameters is more flexible and adaptive than the conventional robust controller with a fixed gain which is derived by the worst-case design for the unknown parameter variations. Moreover, in this chapter we have derived the allowable perturbation region of unknown parameters, and the proposed robust controller can be obtained by solving constrained convex optimization problems. Although the solution of the some matrix inequalities can be applied to the resultant robust controller in the general controller design strategies for the conventional fixed gain robust control, the solutions of the constrained convex optimization problem derived in this chapter cannot be reflected to the resultant robust controller. Note that the proposed controller design strategy includes this

In Section 2 for a class of uncertain linear systems, we have dealt with a design problem of centralized adaptive gain robust state feedback controllers. Although the standard LQ regulator theory for the purpose of generating the desired response is adopted in the existing result [32], the nominal control input is designed by using pole placement constraints. By using the controller gain for the nominal system, the proposed robust control with adjustable timevarying parameter has been designed by solving LMIs. Additionally, based on the derived LMI-based conditions, the constrained convex optimization problem has been obtained for the purpose of the maximization of the allowable perturbation region of uncertainties included in the controlled system. Section 3 extends the result for the centralized adaptive gain robust state feedback controller given in Section 2 to decentralized adaptive gain robust

method is very useful.

4. Conclusions and future works

and so on.

fascinating fact.

Figure 5. Time histories of xi (t) and xið Þt (i = 1, 2). (a) The time histories of x1(t) and x1ð Þt , (b) Time histories of x2(t) and x2ð Þt .

Figure 6. Time histories of x3(t), x3ð Þt , u(t) and u tð Þ. (a) Time histories of x3(t) and x3ð Þt , (b) Time histories of u(t) and u tð Þ.

From these figures, the proposed decentralized adaptive gain robust controller stabilizes the uncertain large-scale interconnected system with system parameters of Eq. (79). Furthermore, one can see that each subsystem achieves good transient behavior close to nominal subsystems by the proposed decentralized robust controller. Thus, the effectiveness of the proposed robust control strategy is shown.

#### 3.4. Summary

In this section, on the basis of the result of Section 2, we have suggested the decentralized adaptive gain robust controller for the large-scale interconnected system with uncertainties. Furthermore, the effectiveness of the proposed controller has been shown via an illustrative example. The proposed adaptive gain robust controller can be easily designed by solving a constrained convex optimization problem and adjust the magnitude of the control input for each subsystem. Therefore, we find that the proposed decentralized robust controller design method is very useful.

Future research subjects include analysis of conservatism for the proposed controller design approach and extension of the proposed adaptive gain robust control strategies to uncertain systems with time delay, decentralized output/observer-based control systems, and so on.
