• Plant rule i for a system:

IF x1ð Þt is Mi<sup>1</sup> and ��� and xnð Þt is Min, THEN

$$\dot{\boldsymbol{x}}(t) = A\_i \boldsymbol{\varkappa}(t) + B\_i \boldsymbol{u}(t),\\ i = \mathbf{1}, \cdots, r. \tag{10}$$

• Control law rule i for a system:

IF x1ð Þt is Mi<sup>1</sup> and ��� and xnð Þt is Min, THEN

$$\mu(t) = -K\_i \mathbf{x}(t), i = \mathbf{1}, \cdots, r. \tag{11}$$

In (10, 11), xið Þt , i ¼ 1, ⋯, n and Mij, i ¼ 1, ⋯, r, j ¼ 1, ⋯, n are state variables and fuzzy sets, respectively, and r is the number of IF-THEN rules. Moreover, Ai ∈ Rn�n, Bi ∈R<sup>n</sup>�p, and Ki ∈ Rp�<sup>n</sup>:

If we follow the usual fuzzy inference method, we can represent the state equations of the TS-type fuzzy model and the TS-type fuzzy control law for a system as follows, respectively [9]:

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) A\_i \mathbf{x}(t) + \sum\_{i=1}^{r} h\_i(\mathbf{x}(t)) B\_i \mathbf{u}(t) \tag{12}$$

and

$$u(t) = -\sum\_{i=1}^{r} h\_i(\varkappa(t)) K\_i \varkappa(t),\tag{13}$$

AiX <sup>þ</sup> XA<sup>T</sup>

AiX <sup>þ</sup> XA<sup>T</sup>

Then the control law

where Ki ≜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>

þ 2∑ r i , j

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

. 0

where Ki ≜4αR�<sup>1</sup>

201

matrix <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 such that

gxt ð Þ ð Þ ≜ ∑<sup>r</sup>

1 2

function

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

BT

j

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

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

� � � <sup>4</sup><sup>α</sup> BiR�<sup>1</sup>

BT <sup>i</sup> X�<sup>1</sup>

x tð Þ<sup>T</sup> <sup>∑</sup> r i¼1 h2

P ¼ ∞ð

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

for all x tð Þ 6¼ 0 and <sup>α</sup>≥1, where Gij <sup>≜</sup> Ai � <sup>2</sup>αBiR�<sup>1</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>

¼ �∑ r i¼1

≜ � ∑ r i¼1

<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>

u∗

BT <sup>i</sup> X�<sup>1</sup>

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

0

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

r i¼1 ∑ r j¼1

Gii � � �

2 � �<sup>T</sup>

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

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

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

hið Þ x tð Þ <sup>4</sup>αR�<sup>1</sup>

hið Þ x tð Þ Kix tð Þ,

<sup>i</sup> þ 2βX , 0, i ¼ 1, ⋯, r,

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

<sup>u</sup><sup>∗</sup>ðÞ¼� <sup>t</sup> <sup>∑</sup>

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

r i¼1

lizing control law for the TS-type fuzzy system in (12) that minimizes the cost

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

hið Þ x tð Þ hjð Þ x tð Þ <sup>X</sup>�<sup>1</sup>

the closed-loop dynamics for the TS-type fuzzy system in (12) with the control law

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.

BT i � � <sup>þ</sup> <sup>2</sup>β<sup>X</sup> , <sup>0</sup>, <sup>1</sup>≤<sup>i</sup> , <sup>j</sup><sup>≤</sup> r,

hið Þ x tð Þ Kix tð Þ, (18)

, i ¼ 1, ⋯, r is the optimal, globally asymptotically stabi-

lxt ð Þþ ð Þ u tð ÞTRu tð Þ h idt, (19)

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

" � �#)x tð Þ

<sup>j</sup> <sup>X</sup>�<sup>1</sup> � � ( )x tð Þ

2

BiR�<sup>1</sup> BT

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

BT <sup>j</sup> X�<sup>1</sup>

x tð Þ becomes

hið Þ x tð Þ Bi

r i¼1

BT <sup>i</sup> <sup>X</sup>�<sup>1</sup> � �x tð Þ

, i ¼ 1, ⋯, r and α ≥1 is a constant. Now, from propositions 1 and 2, assume that there exists a positive definite

� � � � <sup>T</sup>

(17)

(20)

(21)

, and the decay rate of

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

$$h\_i(\mathbf{x}(t)) \triangleq \frac{\prod\_{j=1}^{n} M\_{\vec{\eta}}(\mathbf{x}\_j(t))}{\sum\_{i=1}^{r} M\_{\vec{\eta}}(\mathbf{x}\_j(t))}, \ i = 1, \cdots, r \tag{14}$$

and Mij xjð Þ<sup>t</sup> � � denotes the grade of membership of xjð Þ<sup>t</sup> in the fuzzy set Mij: 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> <sup>i</sup>¼<sup>1</sup>hið Þ¼ x tð Þ 1 for all x tð Þ∈R<sup>n</sup>:
