3.2 Problem definition

The author defines two kinds of problems considered in this section. The first problem is to design a control law for the TS-type fuzzy system in (12) that achieves the global asymptotic stability of the equilibrium point x tðÞ¼ 0 and minimizes the cost function

$$P = \int\_0^\infty \left[ l(\varkappa(t)) + \mathfrak{u}(t)^T R \mathfrak{u}(t) \right] dt,$$

where lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix. The second problem is that the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (12) should be at least β, where β . 0 is a constant.

### 3.3 Optimal control design

We can use the results of propositions 1 and 2 to solve the problems defined in Section 3.2. Specifically, we define f xt ð Þ ð Þ ≜ ∑<sup>r</sup> <sup>i</sup>¼<sup>1</sup>hið Þ x tð Þ Aix tð Þ and gxt ð Þ ð Þ ≜ ∑<sup>r</sup> <sup>i</sup>¼<sup>1</sup>hið Þ x tð Þ Bi for the TS-type fuzzy system in (12). Then, if there exists a radially unbounded and positive definite function Vxt ð Þ ð Þ such that the conditions in (2) and (9) hold, then an optimal control law in the form of (3) for the TS-type fuzzy system in (12) can be found. Thus, we have to construct Vxt ð Þ ð Þ whose time derivative satisfies the conditions in (2, 9). The author uses a quadratic Lyapunov function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ, where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive definite matrix, as the candidate of such a Vxt ð Þ ð Þ , and presents the following theory:

Theorem 1 [62]: For the TS-type fuzzy system in (12), suppose that there exists a function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ with <sup>X</sup> <sup>∈</sup>R<sup>n</sup>�<sup>n</sup> such that

$$X = X^T \succeq \mathbf{0},\tag{15}$$

$$\begin{aligned} &A\_i X + X A\_i^T - 4a B\_i R^{-1} B\_i^T < 0, \quad i = 1, \dots, r, \\ &\frac{1}{2} \left( A\_i X + X A\_i^T + A\_j X + X A\_j^T \right) - 2a \left( B\_i R^{-1} B\_j^T + B\_j R^{-1} B\_i^T \right) < 0, \quad 1 \le i \le j \le r, \end{aligned} \tag{16}$$

and

Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control DOI: http://dx.doi.org/10.5772/intechopen.82181

$$\begin{aligned} &A\_i X + X A\_i^T - 8a B\_i R^{-1} B\_i^T + 2\beta X \le 0, \quad i = 1, \dots, r, \\ &\frac{1}{2} \left( A\_i X + X A\_i^T + A\_j X + X A\_j^T \right) - 4a \left( B\_i R^{-1} B\_j^T + B\_j R^{-1} B\_i^T \right) + 2\beta X \le 0, \quad 1 \le i \le r, \end{aligned} \tag{17}$$

where <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix and <sup>α</sup> <sup>≥</sup>1 and <sup>β</sup> . 0 are constants. Then the control law

$$u^\*(t) = -\sum\_{i=1}^r h\_i(\varkappa(t)) K\_i \varkappa(t),\tag{18}$$

where Ki ≜4αR�<sup>1</sup> BT <sup>i</sup> X�<sup>1</sup> , i ¼ 1, ⋯, r is the optimal, globally asymptotically stabilizing control law for the TS-type fuzzy system in (12) that minimizes the cost function

$$P = \int\_0^\infty \left[ l(\varkappa(t)) + u(t)^T R u(t) \right] dt,\tag{19}$$

where lxt ð Þ ð Þ is given by

x t \_ðÞ¼ ∑ r i¼1

hið Þ x tð Þ ≜

P ¼ ∞ð

Section 3.2. Specifically, we define f xt ð Þ ð Þ ≜ ∑<sup>r</sup>

0

and

Aerospace Engineering

x tð Þ∈R<sup>n</sup>:

cost function

gxt ð Þ ð Þ ≜ ∑<sup>r</sup>

AiX <sup>þ</sup> XAT

AiX <sup>þ</sup> XAT

1 2

200

and

3.2 Problem definition

3.3 Optimal control design

function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup>

function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup>

<sup>i</sup> � <sup>4</sup>αBiR�<sup>1</sup>

� �

BT

j

<sup>i</sup> <sup>þ</sup> AjX <sup>þ</sup> XAT

hið Þ x tð Þ Aix tð Þþ ∑

r i¼1

<sup>j</sup>¼<sup>1</sup> Mij xjð Þ<sup>t</sup> � �

and Mij xjð Þ<sup>t</sup> � � denotes the grade of membership of xjð Þ<sup>t</sup> in the fuzzy set Mij:

The author defines two kinds of problems considered in this section. The first problem is to design a control law for the TS-type fuzzy system in (12) that achieves the global asymptotic stability of the equilibrium point x tðÞ¼ 0 and minimizes the

> lxt ð Þþ ð Þ u tð ÞTRu tð Þ h i

where lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix. The second problem is that the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (12) should be at least β, where β . 0 is a constant.

We can use the results of propositions 1 and 2 to solve the problems defined in

radially unbounded and positive definite function Vxt ð Þ ð Þ such that the conditions in (2) and (9) hold, then an optimal control law in the form of (3) for the TS-type fuzzy system in (12) can be found. Thus, we have to construct Vxt ð Þ ð Þ whose time derivative satisfies the conditions in (2, 9). The author uses a quadratic Lyapunov

Theorem 1 [62]: For the TS-type fuzzy system in (12), suppose that there exists a

BT

<sup>j</sup> <sup>þ</sup> BjR�<sup>1</sup>

� �

x tð Þ with <sup>X</sup> <sup>∈</sup>R<sup>n</sup>�<sup>n</sup> such that

the candidate of such a Vxt ð Þ ð Þ , and presents the following theory:

<sup>i</sup> , 0, i ¼ 1, ⋯, r,

� <sup>2</sup><sup>α</sup> BiR�<sup>1</sup>

<sup>i</sup>¼<sup>1</sup>hið Þ x tð Þ Bi for the TS-type fuzzy system in (12). Then, if there exists a

u tðÞ¼� ∑

where hi, i ¼ 1, ⋯, r are the normalized weight functions given by

Q<sup>n</sup>

∑r

Here,hi, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r in (14) satisfy hið Þ x tð Þ <sup>≥</sup>0, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r and <sup>∑</sup><sup>r</sup>

r i¼1

hið Þ x tð Þ Biu tð Þ (12)

<sup>i</sup>¼<sup>1</sup>hið Þ¼ x tð Þ 1 for all

hið Þ x tð Þ Kix tð Þ, (13)

<sup>i</sup>¼<sup>1</sup>Mij xjð Þ<sup>t</sup> � � , i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r (14)

dt,

<sup>i</sup>¼<sup>1</sup>hið Þ x tð Þ Aix tð Þ and

x tð Þ, where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive definite matrix, as

<sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . <sup>0</sup>, (15)

, 0, 1≤i , j ≤r,

(16)

BT i

$$\begin{split} l(\mathbf{x}(t)) &= -4\alpha^2 \mathbf{x}(t)^T \left\{ \sum\_{i=1}^r h\_i^2(\mathbf{x}(t)) \left( \mathbf{G}\_{ii}^T \mathbf{X}^{-1} + \mathbf{X}^{-1} \mathbf{G}\_{ii} \right) \\ &+ 2 \sum\_{i$$

for all x tð Þ 6¼ 0 and <sup>α</sup>≥1, where Gij <sup>≜</sup> Ai � <sup>2</sup>αBiR�<sup>1</sup> BT <sup>j</sup> X�<sup>1</sup> , and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (12) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (18) is at least <sup>β</sup>, where <sup>β</sup> . 0 is a constant.

Proof: Suppose that Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ, where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive definite matrix. Then, from proposition 1, the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) with gxt ð Þ ð Þ ≜ ∑<sup>r</sup> <sup>i</sup>¼<sup>1</sup>hið Þ x tð Þ Bi and Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ becomes

$$\begin{split} u^\*(t) &= -2aR^{-1} \left[ L\_{\mathbf{g}} V(\mathbf{x}(t)) \right]^T \\ &= -2aR^{-1} \left[ 2\mathbf{x}(t)^T \mathbf{X}^{-1} \left( \sum\_{i=1}^r h\_i(\mathbf{x}(t)) B\_i \right) \right]^T \\ &= -\sum\_{i=1}^r h\_i(\mathbf{x}(t)) \left( 4aR^{-1} B\_i^T \mathbf{X}^{-1} \right) \mathbf{x}(t) \\ &\triangleq -\sum\_{i=1}^r h\_i(\mathbf{x}(t)) K\_i \mathbf{x}(t), \end{split} \tag{21}$$

where Ki ≜4αR�<sup>1</sup> BT <sup>i</sup> X�<sup>1</sup> , i ¼ 1, ⋯, r and α ≥1 is a constant.

Now, from propositions 1 and 2, assume that there exists a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 such that

$$\begin{split} \dot{V}(\mathbf{x}(t)) \Big|\_{\mathbf{x}(t) = \frac{1}{2} \mathbf{x}^\*(t)} &= L\_f V(\mathbf{x}(t)) + \frac{1}{2} \left[ L\_{\tilde{\mathbf{x}}} V(\mathbf{x}(t)) \right] \mathbf{u}^\*(t) \\ &= \mathbf{x}(t)^T \Big{{}^{\tilde{\mathbf{x}}} \Big{{}^{\tilde{\mathbf{x}}}} h\_i(\mathbf{x}(t))^2 \left( \mathbf{G}\_{\tilde{\mathbf{x}}}^T \mathbf{X}^{-1} + \mathbf{X}^{-1} \mathbf{G}\_{\tilde{\mathbf{u}}} \right) \\ &+ 2 \sum\_{i$$

Now, consider the TS-type fuzzy system with a common input matrix, which is

Theorem 2 [62]: For the TS-type fuzzy system in (26), suppose that there exists a

where <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix and <sup>α</sup> <sup>≥</sup>1 and <sup>β</sup> . 0 are constants.

<sup>i</sup> <sup>X</sup>�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup>

for all x tð Þ 6¼ 0 and α ≥1, and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (26) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (30) is at least <sup>β</sup>, where

inite matrix. Then, from proposition 1, the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) with gxt ð Þ ð Þ <sup>≜</sup> <sup>B</sup>

<sup>u</sup><sup>∗</sup>ðÞ¼� <sup>t</sup> <sup>2</sup>αR�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup>

¼ �4αR�<sup>1</sup>

≜ � Kx tð Þ,

¼ �2αR�<sup>1</sup> <sup>2</sup>x tð ÞTX�<sup>1</sup>

BTX�<sup>1</sup>

B h i<sup>T</sup>

x tð Þ

x tð Þ with <sup>X</sup> <sup>∈</sup>Rn�<sup>n</sup> such that

<sup>i</sup> � <sup>4</sup>αBR�<sup>1</sup>

law for the TS-type fuzzy system in (26) that minimizes the cost function

<sup>i</sup> � <sup>8</sup>αBR�<sup>1</sup>

hið Þ x tð Þ Aix tðÞþ Bu tð Þ: (26)

<sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . <sup>0</sup>, (27)

<sup>B</sup><sup>T</sup> , <sup>0</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r, (28)

BT <sup>þ</sup> <sup>2</sup>β<sup>X</sup> , <sup>0</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r, (29)

<sup>u</sup><sup>∗</sup>ðÞ¼� <sup>t</sup> Kx tð Þ, (30)

lxt ð Þþ ð Þ u tð ÞTRu tð Þ h idt, (31)

BR�<sup>1</sup>

x tð Þ, where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive def-

BTX�<sup>1</sup>

(32)

(33)

BTX�<sup>1</sup> is the optimal, globally asymptotically stabilizing control

Ai � � � <sup>4</sup>αX�<sup>1</sup>

� �x tð Þ

x t \_ðÞ¼ ∑ r i¼1

Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control

Then, the author presents the following theory:

DOI: http://dx.doi.org/10.5772/intechopen.82181

AiX <sup>þ</sup> XA<sup>T</sup>

P ¼ ∞ð

x tð Þ<sup>T</sup> <sup>∑</sup> r i¼1

Proof: Suppose that Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup>

x tð Þ becomes

ð Þ <sup>α</sup> � <sup>1</sup> x tð Þ<sup>T</sup> <sup>4</sup>X�<sup>1</sup>

0

hið Þ x tð Þ <sup>A</sup><sup>T</sup>

BR�<sup>1</sup> BTX�<sup>1</sup> � �x tð Þ

AiX <sup>þ</sup> XAT

described by

and

Then the control law:

where K ≜4αR�<sup>1</sup>

where lxt ð Þ ð Þ is given by

lxt ð Þ¼� ð Þ <sup>4</sup>α<sup>2</sup>

<sup>þ</sup> <sup>4</sup>α<sup>2</sup>

. 0

β . 0 is a constant.

203

and Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup>

function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup>

for all x tð Þ 6¼ <sup>0</sup>, where Gij <sup>≜</sup> Ai � <sup>1</sup> <sup>2</sup> Bi <sup>4</sup>αR�<sup>1</sup> BT <sup>j</sup> <sup>X</sup>�<sup>1</sup> � � <sup>¼</sup> Ai � <sup>1</sup> <sup>2</sup> BiKj, and

$$\begin{aligned} \dot{V}(\mathbf{x}(t))|\_{\mathbf{u}(t)=\mathbf{u}^\*(t)} &= L\_f V(\mathbf{x}(t)) + \left[L\_{\overline{g}} V(\mathbf{x}(t))\right] \mathbf{u}^\*(t) \\ &= \mathbf{x}(t)^T \left\{ \sum\_{i=1}^r h\_i^2(\mathbf{x}(t)) \left(Q\_{\overline{u}i}^T \mathbf{X}^{-1} + \mathbf{X}^{-1} Q\_{\overline{u}i}\right) \right. \\ &\left. + 2 \sum\_{i$$

for all x tð Þ, where Qij <sup>≜</sup> Ai � Bi <sup>4</sup>αR�<sup>1</sup> BT <sup>j</sup> <sup>X</sup>�<sup>1</sup> � � <sup>¼</sup> Ai � BiKj and <sup>β</sup> . 0 is a constant. Since the normalized weight functions hi, i ¼ 1, ⋯, r in (22) and (23) satisfy hið Þ x tð Þ hjð Þ x tð Þ <sup>≥</sup>0, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r, j <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r and <sup>∑</sup><sup>r</sup> <sup>i</sup>¼<sup>1</sup>∑<sup>r</sup> <sup>j</sup>¼<sup>1</sup>hið Þ x tð Þ hjð Þ¼ x tð Þ <sup>1</sup> (i.e., ∑<sup>r</sup> <sup>i</sup>¼<sup>1</sup>∑<sup>r</sup> <sup>j</sup>¼<sup>1</sup>hið Þ x tð Þ hjð Þ¼ x tð Þ <sup>∑</sup><sup>r</sup> <sup>i</sup>¼<sup>1</sup>h<sup>2</sup> <sup>i</sup>ð Þþ x tð Þ <sup>2</sup>∑<sup>r</sup> i , j hið Þ x tð Þ hjð Þ¼ x tð Þ 1) for all x tð Þ∈Rn, sufficient conditions for satisfying (22) and (23) are

$$\begin{aligned} &G\_{ii}^T X^{-1} + X^{-1} G\_{ii} < 0, \quad i = 1, \cdots, r, \\ &\left(\frac{G\_{\vec{\eta}} + G\_{\vec{\mu}}}{2}\right)^T X^{-1} + X^{-1} \left(\frac{G\_{\vec{\eta}} + G\_{\vec{\mu}}}{2}\right) < 0, \quad 1 \le i \le j \le r \end{aligned} \tag{24}$$

and

$$\begin{aligned} &Q\_{ii}^T X^{-1} + X^{-1} Q\_{ii} + 2\beta X^{-1} < 0, \; i = 1, \dots, r, \\ &\left(\frac{Q\_{ij} + Q\_{ji}}{2}\right)^T X^{-1} + X^{-1} \left(\frac{Q\_{ji} + Q\_{ji}}{2}\right) + 2\beta X^{-1} < 0, \; 1 \le i < j \le r, \end{aligned} \tag{25}$$

respectively. If all sets of inequalities in (24, 25) are pre- and post-multiplied by a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . <sup>0</sup>, then we can transform the nonlinear conditions in (24, 25) into the linear conditions in (16, 17), respectively. Thus, by the results of propositions 1 and 2, we see that the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (21) with a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 satisfying the conditions in (16, 17) becomes the optimal, globally asymptotically stabilizing control law for the TS-type fuzzy system in (12) that minimizes the cost function in (19), where lxt ð Þ ð Þ in (20) comes from (5) and satisfies lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and α≥ 1 by (22) and the property of <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . <sup>0</sup>, and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (12) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (21) becomes at least <sup>β</sup> . <sup>0</sup>: This completes the proof.

Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control DOI: http://dx.doi.org/10.5772/intechopen.82181

Now, consider the TS-type fuzzy system with a common input matrix, which is described by

$$\dot{\varkappa}(t) = \sum\_{i=1}^{r} h\_i(\varkappa(t)) A\_i \varkappa(t) + Bu(t). \tag{26}$$

Then, the author presents the following theory:

Theorem 2 [62]: For the TS-type fuzzy system in (26), suppose that there exists a function Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ with <sup>X</sup> <sup>∈</sup>Rn�<sup>n</sup> such that

$$X = X^T \succeq \mathbf{0},\tag{27}$$

$$A\_i X + X A\_i^T - 4a B R^{-1} B^T \le 0, \quad i = 1, \dots, r,\tag{28}$$

and

V xt \_ ð Þ ð Þ u tð Þ¼<sup>1</sup>

� �

Aerospace Engineering

V xt \_ ð Þ ð Þ u tð Þ¼u∗ð Þ<sup>t</sup> �

(i.e., ∑<sup>r</sup>

and

<sup>i</sup>¼<sup>1</sup>∑<sup>r</sup>

Q<sup>T</sup>

completes the proof.

202

QijþQji 2 � �<sup>T</sup>

<sup>2</sup> u∗ð Þt

� <sup>¼</sup> LfVxt ð Þþ ð Þ <sup>1</sup>

þ 2 ∑ r i , j

, 0

for all x tð Þ 6¼ <sup>0</sup>, where Gij <sup>≜</sup> Ai � <sup>1</sup>

þ 2∑ r i , j

<sup>¼</sup> x tð Þ<sup>T</sup> <sup>∑</sup>

, � <sup>2</sup>βx tð ÞTX�<sup>1</sup>

for all x tð Þ, where Qij <sup>≜</sup> Ai � Bi <sup>4</sup>αR�<sup>1</sup>

hið Þ x tð Þ hjð Þ x tð Þ <sup>≥</sup>0, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r, j <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r and <sup>∑</sup><sup>r</sup>

<sup>j</sup>¼<sup>1</sup>hið Þ x tð Þ hjð Þ¼ x tð Þ <sup>∑</sup><sup>r</sup>

iiX�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup>

G<sup>T</sup>

iiX�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup>

GijþGji 2 � �<sup>T</sup>

<sup>¼</sup> x tð Þ<sup>T</sup> <sup>∑</sup>

r i¼1

� <sup>¼</sup> LfVxt ð Þþ ð Þ LgVxt ð Þ ð Þ � �u<sup>∗</sup>ð Þ<sup>t</sup>

r i¼1 h2 2

hið Þ x tð Þ <sup>2</sup> <sup>G</sup><sup>T</sup>

<sup>i</sup>ð Þ x tð Þ <sup>Q</sup><sup>T</sup>

x tð Þ

<sup>i</sup>¼<sup>1</sup>h<sup>2</sup>

<sup>X</sup>�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup> Gij <sup>þ</sup> Gji

Qii <sup>þ</sup> <sup>2</sup>βX�<sup>1</sup> , <sup>0</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r,

2 � �

x tð Þ∈Rn, sufficient conditions for satisfying (22) and (23) are

<sup>X</sup>�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup> Qij <sup>þ</sup> Qji

hið Þ x tð Þ hjð Þ x tð Þ Qij <sup>þ</sup> Qji

hið Þ x tð Þ hjð Þ x tð Þ Gij <sup>þ</sup> Gji

LgVxt ð Þ ð Þ � �u<sup>∗</sup>

� � �

<sup>2</sup> Bi <sup>4</sup>αR�<sup>1</sup>

� � �

iiX�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup>

2 � �<sup>T</sup>

BT <sup>j</sup> <sup>X</sup>�<sup>1</sup> � �

<sup>i</sup>ð Þþ x tð Þ <sup>2</sup>∑<sup>r</sup>

<sup>i</sup>¼<sup>1</sup>∑<sup>r</sup>

i , j

, 0, 1≤i , j≤r

<sup>þ</sup> <sup>2</sup>βX�<sup>1</sup> , <sup>0</sup>, <sup>1</sup>≤<sup>i</sup> , <sup>j</sup><sup>≤</sup> r,

stant. Since the normalized weight functions hi, i ¼ 1, ⋯, r in (22) and (23) satisfy

Gii , 0, i ¼ 1, ⋯, r,

2 � �

respectively. If all sets of inequalities in (24, 25) are pre- and post-multiplied by a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . <sup>0</sup>, then we can transform the nonlinear conditions in (24, 25) into the linear conditions in (16, 17), respectively. Thus, by the results of propositions 1 and 2, we see that the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (21) with a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 satisfying the conditions in (16, 17) becomes the optimal, globally asymptotically stabilizing control law for the TS-type fuzzy system in (12) that minimizes the cost function in (19), where lxt ð Þ ð Þ in (20) comes from (5) and satisfies lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and α≥ 1 by (22) and the property of <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . <sup>0</sup>, and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (12) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (21) becomes at least <sup>β</sup> . <sup>0</sup>: This

ð Þt

Gii

<sup>X</sup>�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup> Gij <sup>þ</sup> Gji

" � �#)

<sup>¼</sup> Ai � <sup>1</sup>

" � �#)

<sup>X</sup>�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup> Qij <sup>þ</sup> Qji

¼ Ai � BiKj and β . 0 is a con-

hið Þ x tð Þ hjð Þ¼ x tð Þ 1) for all

<sup>j</sup>¼<sup>1</sup>hið Þ x tð Þ hjð Þ¼ x tð Þ <sup>1</sup>

2

<sup>2</sup> BiKj, and

2

x tð Þ

(22)

x tð Þ

(23)

(24)

(25)

iiX�<sup>1</sup> <sup>þ</sup> <sup>X</sup>�<sup>1</sup>

2 � �<sup>T</sup>

BT <sup>j</sup> <sup>X</sup>�<sup>1</sup> � �

Qii

$$A\_i X + X A\_i^T - 8aBR^{-1}B^T + 2\beta X \le 0, \ i = 1, \cdots, r,\tag{29}$$

where <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix and <sup>α</sup> <sup>≥</sup>1 and <sup>β</sup> . 0 are constants. Then the control law:

$$
\mu^\*(t) = -K\mathbf{x}(t),\tag{30}
$$

where K ≜4αR�<sup>1</sup> BTX�<sup>1</sup> is the optimal, globally asymptotically stabilizing control law for the TS-type fuzzy system in (26) that minimizes the cost function

$$P = \int\_0^\infty \left[ l(\varkappa(t)) + u(t)^T R u(t) \right] dt,\tag{31}$$

where lxt ð Þ ð Þ is given by

$$\begin{split} l(\mathbf{x}(t)) &= -4a^2 \mathbf{x}(t)^T \left\{ \sum\_{i=1}^r h\_i(\mathbf{x}(t)) \left( A\_i^T X^{-1} + X^{-1} A\_i \right) - 4a X^{-1} \mathbf{B} \mathbf{R}^{-1} \mathbf{B}^T X^{-1} \right\} \mathbf{x}(t) \\ &+ 4a^2 (a-1) \mathbf{x}(t)^T \left( 4X^{-1} \mathbf{B} \mathbf{R}^{-1} \mathbf{B}^T X^{-1} \right) \mathbf{x}(t) \\ &> 0 \end{split} \tag{32}$$

for all x tð Þ 6¼ 0 and α ≥1, and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (26) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (30) is at least <sup>β</sup>, where β . 0 is a constant.

Proof: Suppose that Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ, where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive definite matrix. Then, from proposition 1, the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) with gxt ð Þ ð Þ <sup>≜</sup> <sup>B</sup> and Vxt ð Þ¼ ð Þ x tð ÞTX�<sup>1</sup> x tð Þ becomes

$$\begin{aligned} \boldsymbol{u}^\*(t) &= -2a\boldsymbol{R}^{-1} \left[ \boldsymbol{L}\_\sharp \boldsymbol{V}(\boldsymbol{\varkappa}(t)) \right]^T \\ &= -2a\boldsymbol{R}^{-1} \left[ 2\boldsymbol{\varkappa}(t)^T \boldsymbol{X}^{-1} \boldsymbol{B} \right]^T \\ &= -4a\boldsymbol{R}^{-1} \boldsymbol{B}^T \boldsymbol{X}^{-1} \boldsymbol{\varkappa}(t) \\ &\triangleq -K\boldsymbol{\varkappa}(t), \end{aligned} \tag{33}$$

where K ≜4αR�<sup>1</sup> BTX�<sup>1</sup> and α≥1 is a constant.

Now, from propositions 1 and 2, assume that there exists a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 such that

4.1 Rigid body model

DOI: http://dx.doi.org/10.5772/intechopen.82181

and has the property of

q1ðÞ¼ t

205

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi <sup>1</sup> � qvð Þ<sup>t</sup> � � � � 2 2

4.2 Optimal control design

as follows [36]:

First, the dynamic equation of the rotational motion of a rigid body is described

Ωð Þ ωð Þt JωðÞþt J

where <sup>ω</sup>ðÞ¼ <sup>t</sup> ½ � <sup>ω</sup>1ð Þ<sup>t</sup> <sup>ω</sup>2ð Þ<sup>t</sup> <sup>ω</sup>3ð Þ<sup>t</sup> <sup>T</sup> <sup>∈</sup>R<sup>3</sup> is the angular velocity vector of the body in the body-fixed frame, u tðÞ¼ ½ � <sup>u</sup>1ð Þ<sup>t</sup> <sup>u</sup>2ð Þ<sup>t</sup> <sup>u</sup>3ð Þ<sup>t</sup> <sup>T</sup> <sup>∈</sup>R<sup>3</sup> is the control torque vector

<sup>Ω</sup>ð Þ <sup>ω</sup>ð Þ<sup>t</sup> <sup>T</sup>ωðÞ� <sup>t</sup> <sup>0</sup>, <sup>∀</sup>ωð Þ<sup>t</sup> <sup>∈</sup> <sup>R</sup><sup>3</sup>

q t \_ðÞ¼ <sup>1</sup> 2

Fqt ð Þ ð Þ : <sup>R</sup><sup>4</sup> ! <sup>R</sup><sup>4</sup>�<sup>3</sup> denotes the kinematics Jacobian matrix defined as

describes all eigenaxis rotations [36]. Thus, we can write q1ð Þt as

for all �π rad≤ϕ ≤π rad:.

kinematics system in (41). This observation gives the following theorem:

input, and let the control law for the kinematics system in (41) be

Fqt ð Þ ð Þ <sup>≜</sup> �qvð Þ<sup>t</sup> <sup>T</sup>

where q tðÞ¼ <sup>q</sup>1ð Þ<sup>t</sup> <sup>q</sup>2ð Þ<sup>t</sup> <sup>q</sup>3ð Þ<sup>t</sup> <sup>q</sup>4ð Þ<sup>t</sup> � �<sup>T</sup> <sup>≜</sup> <sup>q</sup>1ð Þ<sup>t</sup> qvð Þ<sup>t</sup> <sup>T</sup> h i<sup>T</sup>

where I<sup>3</sup>�<sup>3</sup> denotes the 3 � 3 identity matrix.

Second, the kinematic equation of rotational motion of a rigid body described in

<sup>q</sup>1ð Þ<sup>t</sup> <sup>I</sup><sup>3</sup>�<sup>3</sup> <sup>þ</sup> <sup>Ω</sup> qvð Þ<sup>t</sup> � �<sup>T</sup>

With the notations of the Euler axis ^e ∈R<sup>3</sup> and Euler angle ϕ∈R, we define the quaternion by q1ð Þt ≜ cosð Þ ϕ=2 and qvð Þt ≜^e sin ð Þ ϕ=2 : The quaternion q tð Þ is subject to the unit length constraint of k k q tð Þ <sup>2</sup> ¼ 1 for all t≥ 0 and is a kinematic parameter set that can represent the orientation of a body and [36]. From the definition of the quaternion, we see that q1ð Þt satisfies q1ð Þt ≥ 0 for all �π rad≤ ϕ≤π rad, which

First, it is observed that two state equations given by (38) and (41) represent a system in cascade interconnection. That is, the angular velocity vector indirectly controls the kinematics system in (41). Thus, the angular velocity vector can be regarded as a virtual control input of the kinematics system in (41) to stabilize the

Theorem 3 [62]: Consider the kinematics system in (41) with ωð Þt to be the control

" #

0 ω3ðÞ � t ω2ð Þt �ω3ð Þt 0 ω1ð Þt ω2ðÞ � t ω1ð Þt 0

�1

u tð Þ, (38)

: (39)

∈ R<sup>4</sup> is the quaternion and

, (41)

3 7 5

Fqt ð Þ ð Þ ωð Þt , (40)

<sup>T</sup> . 0:

ω\_ðÞ¼ t J

Ωð Þ ωð Þt ≜

terms of the quaternion is given as follows [36]:

�1

Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control

And <sup>Ω</sup>ð Þ <sup>ω</sup>ð Þ<sup>t</sup> <sup>∈</sup>R3�<sup>3</sup> denotes a skew-symmetric matrix defined by

2 6 4

of the body, and <sup>J</sup> <sup>∈</sup>R3�<sup>3</sup> is the inertia matrix of the body and satisfies <sup>J</sup> <sup>¼</sup> <sup>J</sup>

$$\begin{aligned} \dot{V}(\mathbf{x}(t)) \Big|\_{\mathbf{x}(t) = \frac{1}{2}\mathbf{u}^\*(t)} &= L\_f V(\mathbf{x}(t)) + \frac{1}{2} \left[ L\_\xi V(\mathbf{x}(t)) \right] \mathbf{u}^\*(t) \\ &= \mathbf{x}(t)^T \left\{ \sum\_{i=1}^r h\_i(\mathbf{x}(t)) \left( A\_i^T X^{-1} + X^{-1} A\_i \right) - 4a X^{-1} B R^{-1} B^T X^{-1} \right\} \mathbf{x}(t) \\ &< 0 \end{aligned} \tag{34}$$

for all x tð Þ 6¼ 0 and

$$\begin{split} \dot{V}(\mathbf{x}(t))|\_{\mathbf{u}(t)=\mathbf{u}^\*(t)} &= L\_f V(\mathbf{x}(t)) + \left[ L\_\mathcal{E} V(\mathbf{x}(t)) \right] \mathbf{u}^\*(t) \\ &= \mathbf{x}(t)^T \left\{ \sum\_{i=1}^r h\_i(\mathbf{x}(t)) \left( A\_i^T X^{-1} + X^{-1} A\_i \right) - 8\alpha X^{-1} B R^{-1} B^T X^{-1} \right\} \mathbf{x}(t) \\ &< -2\beta \mathbf{x}(t)^T X^{-1} \mathbf{x}(t) \end{split} \tag{35}$$

for all x tð Þ, where β . 0 is a constant. Since the normalized weight functions hi, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r in (34, 35) satisfy hið Þ x tð Þ <sup>≥</sup>0, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r and <sup>∑</sup><sup>r</sup> <sup>i</sup>¼<sup>1</sup>hið Þ¼ x tð Þ 1 for all x tð Þ∈Rn, sufficient conditions for satisfying (34, 35) are

$$A\_i^T X^{-1} + X^{-1} A\_i - 4aX^{-1}BR^{-1}B^T X^{-1} \le 0, \quad i = 1, \dots, r \tag{36}$$

and

$$A\_i^T X^{-1} + X^{-1} A\_i - 8aX^{-1} \mathcal{B} \mathcal{R}^{-1} \mathcal{B}^T X^{-1} + 2\mathcal{\rho} \mathcal{X}^{-1} \le 0, \ i = 1, \cdots, r,\tag{37}$$

respectively. If all sets of inequalities in (36, 37) are pre- and post-multiplied by a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . <sup>0</sup>, then we can transform the nonlinear conditions in (36, 37) into the linear conditions in (28, 29), respectively. Therefore, by the results of propositions 1 and 2, we see that the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (33) with a positive definite matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 satisfying the conditions in (28, 29) becomes the optimal, globally asymptotically stabilizing control law for the TS-type fuzzy system in (26) that minimizes the cost function in (31), where lxt ð Þ ð Þ in (32) comes from (5) and satisfies lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and α≥ 1 by (34) and the property of <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . <sup>0</sup>, and the decay rate of the closed-loop dynamics for the TS-type fuzzy system in (26) with the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (33) becomes at least <sup>β</sup> . <sup>0</sup>: This completes the proof.

Note that the problem appearing in Theorems 1 and 2 is to find a matrix X ∈ Rn�<sup>n</sup> subject to some linear constraints in the form of linear matrix inequality (LMI). Therefore, this problem is an LMI-based problem [32], and we can efficiently solve the LMI-based problem by the LMI Control Toolbox of MATLAB [35]. In this chapter, the author uses the LMI Control Toolbox of MATLAB [35] as the solver for the LMI-based problem in the optimal control design.
