4.1 Rigid body model

where K ≜4αR�<sup>1</sup>

Aerospace Engineering

Vxt \_ ð Þ ð Þ u tð Þ¼<sup>1</sup>

 

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

<sup>2</sup> u∗ð Þt

for all x tð Þ 6¼ 0 and

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

and

AT

completes the proof.

204

4. A control design example

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

, 0

AT

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

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

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

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

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

Ai � <sup>8</sup>αX�<sup>1</sup>

the LMI-based problem in the optimal control design.

r i¼1

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

r i¼1

BTX�<sup>1</sup> and α≥1 is a constant.

2

hið Þ x tð Þ <sup>A</sup><sup>T</sup>

hið Þ x tð Þ AT

x tð Þ

Ai � <sup>4</sup>αX�<sup>1</sup>

BR�<sup>1</sup>

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>

x tð Þ∈Rn, sufficient conditions for satisfying (34, 35) are

Now, from propositions 1 and 2, assume that there exists a positive definite

LgVxt ð Þ ð Þ <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup>

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

ð Þt

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

for all x tð Þ, where β . 0 is a constant. Since the normalized weight functions

BR�<sup>1</sup>

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

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

As a control design example, the author considers the three-axis attitude stabili-

zation problem of a rigid body and illustrates the effectiveness of the optimal

control design for TS-type fuzzy systems presented in Section 3.

Ai � <sup>4</sup>αX�<sup>1</sup>

Ai � <sup>8</sup>αX�<sup>1</sup>

BR�<sup>1</sup>

BR�<sup>1</sup>

<sup>B</sup>TX�<sup>1</sup> , <sup>0</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r (36)

BTX�<sup>1</sup> <sup>þ</sup> <sup>2</sup>βX�<sup>1</sup> , <sup>0</sup>, i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, r, (37)

BTX�<sup>1</sup>

BTX�<sup>1</sup>

x tð Þ

x tð Þ

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

(34)

(35)

First, the dynamic equation of the rotational motion of a rigid body is described as follows [36]:

$$
\dot{o}(t) = f^{-1} \Omega(o(t)) J o(t) + f^{-1} u(t), \tag{38}
$$

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 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> <sup>T</sup> . 0: 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

$$\boldsymbol{\Omega}(\boldsymbol{\alpha}(t)) \triangleq \begin{bmatrix} \mathbf{0} & \boldsymbol{\alpha}\_{3}(t) & -\boldsymbol{\alpha}\_{2}(t) \\ -\boldsymbol{\alpha}\_{3}(t) & \mathbf{0} & \boldsymbol{\alpha}\_{1}(t) \\ \boldsymbol{\alpha}\_{2}(t) & -\boldsymbol{\alpha}\_{1}(t) & \mathbf{0} \end{bmatrix}.$$

and has the property of

$$\left(\Omega(\boldsymbol{\alpha}(t))\right)^{T}\boldsymbol{\alpha}(t) \equiv \mathbf{0}, \ \forall \boldsymbol{\alpha}(t) \in \boldsymbol{R}^{3}. \tag{39}$$

Second, the kinematic equation of rotational motion of a rigid body described in terms of the quaternion is given as follows [36]:

$$
\dot{q}(t) = \frac{1}{2} F(q(t)) \alpha(t),
\tag{40}
$$

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> ∈ R<sup>4</sup> is the quaternion and 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

$$F(q(t)) \triangleq \begin{bmatrix} -q\_v(t)^T \\ q\_1(t)I\_{3 \times 3} + \mathfrak{Q}\left(q\_v(t)\right)^T \end{bmatrix},\tag{41}$$

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

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

$$q\_1(t) = \sqrt{1 - \left\| q\_v(t) \right\|\_2^2} \text{ for all } -\pi \text{ rad} \le \phi \le \pi \text{ rad}... $$

#### 4.2 Optimal control design

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 kinematics system in (41). This observation gives the following theorem: Theorem 3 [62]: Consider the kinematics system in (41) with ωð Þt to be the control input, and let the control law for the kinematics system in (41) be

$$a\_{\nu}(t) = -k\_1 q\_{\nu}(t),\tag{42}$$

<sup>B</sup> <sup>≜</sup> <sup>J</sup>

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

<sup>J</sup> <sup>¼</sup> diag 10½ � ; <sup>15</sup>; <sup>20</sup> kg � <sup>m</sup><sup>2</sup>

∈½ � �0:5; 0:5 , i ¼ 1, ⋯, 3: If we do sampling Axt ð Þ ð Þ in (46) at nine operating

x t \_ðÞ¼ Aix tð Þþ Bu tð Þ, i ¼ 1, ⋯, 9: (48)

h i <sup>¼</sup> ½ � 0 0 , ½ � 0 0:<sup>5</sup> , ½ � <sup>0</sup> � <sup>0</sup>:<sup>5</sup> , ½ � <sup>0</sup>:25 0 , ½ � <sup>0</sup>:25 0:<sup>5</sup> , ½ � �0:25 0 ,

½ � �0:25 � 0:5 , ½ � 0:5 0:5 , ½ � �0:5 � 0:5 , i ¼ 1, ⋯, 3 with the given k<sup>1</sup> ¼ 0:2 and J in (48), we can obtain the following TS-type fuzzy model for the system in (38):

In (49), x tð Þ∈R<sup>6</sup> is the state vector, u tð Þ∈R<sup>3</sup> is the control input vector, Mij, i ¼ 1, ⋯, 9, j ¼ 1, ⋯, 6 are the fuzzy sets defined as in Figures 1 and 2,

Fuzzy sets Mij, i ¼ 1, ⋯, 9, j ¼ 1, ⋯, 3 used to design TS-type fuzzy model in (49) for system in (45).

Ai, i ¼ 1, ⋯, 9 are obtained by sampling Axt ð Þ ð Þ in (46) at the given nine operatingpoints, and B is given in (47). With the normalized weight functions hi, i ¼ 1, ⋯, 9

As a numerical example, the author assumes that

where diag means the diagonal matrix, k<sup>1</sup> ¼ 0:2, and

xδi , xqvi

points of x<sup>δ</sup><sup>i</sup>

defined by

Figure 1.

207

; xqvi

• Plant rule i for the system in (45):

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

IF x1ð Þt is Mi<sup>1</sup> and ��� and x6ð Þt is Mi6, THEN

�1 03�<sup>3</sup>

: (46)

, (47)

" #

where k<sup>1</sup> . 0 is a constant. Then, ωvð Þt in (43) is the global asymptotic stabilizing control law for the kinematics system in (41).

Proof: With the control law ωvð Þt in (43), the closed-loop system of the kinematics system in (41) becomes

$$\dot{q}(t) = -\frac{1}{2}k\_1 F(q(t))q\_v(t) = -\frac{1}{2}k\_1 \begin{bmatrix} -q\_v(t) \\ q\_1(t)I\_{3 \times 3} + \Omega \left(q\_v(t)\right)^T \end{bmatrix} q\_v(t). \tag{43}$$

Now, consider the Lyapunov function candidate Vqt ð Þ¼ ð Þ <sup>q</sup>1ðÞ�<sup>t</sup> <sup>1</sup> � �<sup>2</sup> þ qvð Þ<sup>t</sup> � � � � 2 2 . Taking the time derivative of Vqt ð Þ ð Þ along a nonzero trajectory of the closed-loop system in (44) and using the property of <sup>Ω</sup> qvð Þ<sup>t</sup> � �qvðÞ� <sup>t</sup> 0 for all qvð Þ<sup>t</sup> <sup>∈</sup>R<sup>3</sup> , then the following condition holds:

$$\begin{aligned} \dot{V}(q(t)) &= 2(q\_1(t) - 1)\dot{q}\_1(t) + 2q\_v(t)^T \dot{q}\_v(t) \\ &= 2q\_1(t) \left(\frac{1}{2}k\_1 q\_v(t)^T q\_v(t)\right) - 2\left(\frac{1}{2}k\_1 q\_v(t)^T q\_v(t)\right) \\ &+ 2q\_v(t)^T \left(-\frac{1}{2}k\_1 q\_1(t) q\_v(t) + \frac{1}{2}k\_1 \Omega(q\_v(t)) q\_v(t)\right) \\ &= -k\_1 q\_v(t)^T q\_v(t) < 0 \end{aligned}$$

for all qvð Þt 6¼ 0 and all k<sup>1</sup> . 0: Then, global asymptotic stability of the closedloop dynamics in (44) follows from the Barbashin-Krasovskii theorem [4]. This completes the proof.

Next, we have to stabilize the dynamics system in (38) with making ωð Þt in (38) follow ωvð Þt in (43), and this is a backstepping problem [39]. For solving this problem, the author defines the new variable δð Þt as δð Þt ≜ωð Þ�t ωvðÞ¼ t ωð Þþt k1qvð Þt : For convenience of notation, the author defines x1ð Þt ≜ δ1ð Þt , x2ð Þt ≜δ2ð Þt , x3ð Þt ≜δ3ð Þt , x4ð Þt ≜ q2ð Þt , x5ð Þt ≜q3ð Þt , x6ð Þt ≜q4ð Þt , xδð Þt ≜½x1ð Þt <sup>x</sup>2ð Þ<sup>t</sup> <sup>x</sup>3ð Þ� <sup>t</sup> T, and xqv ð Þ<sup>t</sup> <sup>≜</sup>½ � <sup>x</sup>4ð Þ<sup>t</sup> <sup>x</sup>5ð Þ<sup>t</sup> <sup>x</sup>6ð Þ<sup>t</sup> <sup>T</sup>: Then, with x tð Þ<sup>≜</sup> <sup>x</sup>δð Þ<sup>t</sup> <sup>T</sup> xqv ð Þ<sup>t</sup> <sup>T</sup> h i<sup>T</sup> and q1ðÞ¼ t ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � � 2 2 q for all �π rad≤ ϕ≤π rad,the author represents the state equation for x tð Þ by

$$
\dot{\varkappa}(t) = A(\varkappa(t))\varkappa(t) + Bu(t), \tag{44}
$$

where Axt ð Þ ð Þ <sup>∈</sup>R<sup>6</sup>�<sup>6</sup> and <sup>B</sup>∈R<sup>6</sup>�<sup>3</sup> are defined in (46) and (47), respectively. In (47), 03�<sup>3</sup> denotes the 3 � 3 zero matrix:

Axt ð Þ ð Þ ≜ � J �1 Ω xδð Þ�t k1xqv ð Þ<sup>t</sup> � �<sup>J</sup> þ 1 2 k1Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup> þ 1 2 k1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2 � � q I<sup>3</sup>�<sup>3</sup> � , � �k<sup>1</sup> J �1 Ω xδðÞ�t k1xqv ð Þ<sup>t</sup> � �<sup>J</sup> � 1 2 k2 <sup>1</sup>Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup> � 1 2 k2 1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2 � � q I<sup>3</sup>�<sup>3</sup> � 1 2 Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup> � þ 1 2 k1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2 � � q I<sup>3</sup>�<sup>3</sup> � , � 1 2 k1Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup> � � 1 2 k1 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2 � � q I<sup>3</sup>�<sup>3</sup> � 2 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 5 : (45)

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

$$B \triangleq \begin{bmatrix} J^{-1} \\ \mathbf{0}\_{3 \times 3} \end{bmatrix}. \tag{46}$$

As a numerical example, the author assumes that

$$J = \text{diag}[10, 15, 20] \text{ kg} \cdot \text{m}^2,\tag{47}$$

where diag means the diagonal matrix, k<sup>1</sup> ¼ 0:2, and xδi , xqvi ∈½ � �0:5; 0:5 , i ¼ 1, ⋯, 3: If we do sampling Axt ð Þ ð Þ in (46) at nine operating points of x<sup>δ</sup><sup>i</sup> ; xqvi h i <sup>¼</sup> ½ � 0 0 , ½ � 0 0:<sup>5</sup> , ½ � <sup>0</sup> � <sup>0</sup>:<sup>5</sup> , ½ � <sup>0</sup>:25 0 , ½ � <sup>0</sup>:25 0:<sup>5</sup> , ½ � �0:25 0 , ½ � �0:25 � 0:5 , ½ � 0:5 0:5 , ½ � �0:5 � 0:5 , i ¼ 1, ⋯, 3 with the given k<sup>1</sup> ¼ 0:2 and J in (48), we can obtain the following TS-type fuzzy model for the system in (38):

• Plant rule i for the system in (45):

ωvðÞ¼� t k1qvð Þt , (42)

qvð Þt : (43)

þ

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

q\_ <sup>v</sup>ð Þt

<sup>k</sup>1qvð Þ<sup>t</sup> <sup>T</sup>

� �

<sup>k</sup>1<sup>Ω</sup> qvð Þ<sup>t</sup> � �qvð Þ<sup>t</sup>

qvð Þt

xqv ð Þ<sup>t</sup> <sup>T</sup> h i<sup>T</sup>

and

� <sup>2</sup> <sup>1</sup> 2

> 1 2

� �

where k<sup>1</sup> . 0 is a constant. Then, ωvð Þt in (43) is the global asymptotic stabilizing

Proof: With the control law ωvð Þt in (43), the closed-loop system of the kinemat-

. Taking the time derivative of Vqt ð Þ ð Þ along a nonzero trajectory of the

qvð Þt

k1q1ð Þt qvðÞþt

for all qvð Þt 6¼ 0 and all k<sup>1</sup> . 0: Then, global asymptotic stability of the closedloop dynamics in (44) follows from the Barbashin-Krasovskii theorem [4]. This

Next, we have to stabilize the dynamics system in (38) with making ωð Þt in (38)

ð Þ<sup>t</sup> <sup>≜</sup>½ � <sup>x</sup>4ð Þ<sup>t</sup> <sup>x</sup>5ð Þ<sup>t</sup> <sup>x</sup>6ð Þ<sup>t</sup> <sup>T</sup>: Then, with x tð Þ<sup>≜</sup> <sup>x</sup>δð Þ<sup>t</sup> <sup>T</sup>

where Axt ð Þ ð Þ <sup>∈</sup>R<sup>6</sup>�<sup>6</sup> and <sup>B</sup>∈R<sup>6</sup>�<sup>3</sup> are defined in (46) and (47), respectively. In

� �k<sup>1</sup> J �1

� 1 2 k2 <sup>1</sup>Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup>

� 1 2 k2 1

> � 1 2 k1Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup>

� 1 2 k1

�

for all �π rad≤ ϕ≤π rad,the author represents the state

x t \_ðÞ¼ Axt ð Þ ð Þ x tðÞþ Bu tð Þ, (44)

Ω xδðÞ�t k1xqv ð Þ<sup>t</sup> � �<sup>J</sup>

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2

� � q

I<sup>3</sup>�<sup>3</sup> �

: (45)

I<sup>3</sup>�<sup>3</sup> �

� � q

1 2 k1

Now, consider the Lyapunov function candidate Vqt ð Þ¼ ð Þ <sup>q</sup>1ðÞ�<sup>t</sup> <sup>1</sup> � �<sup>2</sup>

closed-loop system in (44) and using the property of <sup>Ω</sup> qvð Þ<sup>t</sup> � �qvðÞ� <sup>t</sup> 0 for all

<sup>k</sup>1qvð Þ<sup>t</sup> <sup>T</sup>

2

� �

qvð Þt , 0

follow ωvð Þt in (43), and this is a backstepping problem [39]. For solving this problem, the author defines the new variable δð Þt as δð Þt ≜ωð Þ�t ωvðÞ¼ t ωð Þþt k1qvð Þt : For convenience of notation, the author defines x1ð Þt ≜ δ1ð Þt , x2ð Þt ≜δ2ð Þt , x3ð Þt ≜δ3ð Þt , x4ð Þt ≜ q2ð Þt , x5ð Þt ≜q3ð Þt , x6ð Þt ≜q4ð Þt , xδð Þt ≜½x1ð Þt

control law for the kinematics system in (41).

k1Fqt ð Þ ð Þ qvðÞ¼� t

, then the following condition holds:

¼ 2q1ð Þt

<sup>þ</sup> <sup>2</sup>qvð Þ<sup>t</sup> <sup>T</sup> � <sup>1</sup>

¼ �k1qvð Þ<sup>t</sup> <sup>T</sup>

V qt \_ ð Þ¼ ð Þ <sup>2</sup> <sup>q</sup>1ðÞ�<sup>t</sup> <sup>1</sup> � �q\_ <sup>1</sup>ðÞþ<sup>t</sup> <sup>2</sup>qvð Þ<sup>t</sup> <sup>T</sup>

1 2

ics system in (41) becomes

Aerospace Engineering

q t \_ðÞ¼� <sup>1</sup>

qvð Þ<sup>t</sup> � � � � 2 2

qvð Þ<sup>t</sup> <sup>∈</sup>R<sup>3</sup>

completes the proof.

<sup>x</sup>2ð Þ<sup>t</sup> <sup>x</sup>3ð Þ� <sup>t</sup> T, and xqv

equation for x tð Þ by

Axt ð Þ ð Þ ≜

q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � � 2 2

(47), 03�<sup>3</sup> denotes the 3 � 3 zero matrix:

Ω xδð Þ�t k1xqv ð Þ<sup>t</sup> � �<sup>J</sup>

� � q

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 � xqv ð Þ<sup>t</sup> � � � �2 2

� � q

I<sup>3</sup>�<sup>3</sup> � ,

I<sup>3</sup>�<sup>3</sup> � ,

� J �1

þ 1 2 k1Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup>

þ 1 2 k1

�

þ 1 2 k1

1 2 Ω xqv ð Þ<sup>t</sup> � �<sup>T</sup>

q1ðÞ¼ t

206

2

IF x1ð Þt is Mi<sup>1</sup> and ��� and x6ð Þt is Mi6, THEN

$$\dot{\boldsymbol{x}}(t) = A\_i \boldsymbol{x}(t) + Bu(t), \boldsymbol{i} = \mathbf{1}, \cdots, \mathbf{9}. \tag{48}$$

In (49), x tð Þ∈R<sup>6</sup> is the state vector, u tð Þ∈R<sup>3</sup> is the control input vector, Mij, i ¼ 1, ⋯, 9, j ¼ 1, ⋯, 6 are the fuzzy sets defined as in Figures 1 and 2, Ai, i ¼ 1, ⋯, 9 are obtained by sampling Axt ð Þ ð Þ in (46) at the given nine operatingpoints, and B is given in (47). With the normalized weight functions hi, i ¼ 1, ⋯, 9 defined by

Figure 1. Fuzzy sets Mij, i ¼ 1, ⋯, 9, j ¼ 1, ⋯, 3 used to design TS-type fuzzy model in (49) for system in (45).

Figure 2.

Fuzzy sets Mij, i ¼ 1, ⋯, 9, j ¼ 4, ⋯, 6 used to design TS-type fuzzy model in (49) for system in (45).

$$h\_i(\boldsymbol{\kappa}(t)) \triangleq \frac{\prod\_{j=1}^6 M\_{\boldsymbol{\circ}}(\boldsymbol{\kappa}\_j(t))}{\sum\_{i=1}^9 \prod\_{j=1}^6 M\_{\boldsymbol{\circ}}(\boldsymbol{\kappa}\_j(t))}, i = 1, \dots, 9,$$

the TS-type fuzzy model in (49) can be transformed into the following TS-type fuzzy system:

$$\dot{\boldsymbol{x}}(t) = \sum\_{i=1}^{9} h\_i(\boldsymbol{\kappa}(t)) \boldsymbol{A}\_i \boldsymbol{\kappa}(t) + \boldsymbol{B} \boldsymbol{u}(t). \tag{49}$$

and (28) for the TS-type fuzzy system in (50) with α ¼ 2 and R ¼ I3�3, and one is

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

28:0708 2:5099 �1:9744 3:5899 0:3747 �0:1420 1:6733 33:7936 �4:9727 1:3456 1:7520 0:3518 �0:9872 �3:7295 42:8319 1:4713 1:3202 �0:2517

29:1798 18:4270 18:9874 0:6854 0:2322 �1:5053 12:2846 40:0169 18:1900 2:3547 �1:7662 �1:3049 9:4937 13:6425 49:3411 2:4969 0:9770 �4:9648

As the numerical simulation model, the author uses the equations of rotational motion of a rigid body given by (38) and (41), where the inertia matrix is given by (48). The author assumes that the initial conditions at the initial time t<sup>0</sup> ¼ 0 sec for the Euler axis ^<sup>e</sup> and the Euler angle <sup>ϕ</sup> are ^e tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:4896 0:2030 0:<sup>8480</sup> <sup>T</sup> and ϕð Þ¼ t<sup>0</sup> 2:4648 rad, respectively, which give q tð Þ¼ <sup>0</sup> ½0:3320 0:4618 0:1915

<sup>T</sup>: Note that the given <sup>ϕ</sup>ð Þ <sup>t</sup><sup>0</sup> represents an almost upside-down initial orientation of a rigid body. Also, the author assumes a rest-to-rest maneuver of a rigid body, and, thus, the author assumes that the initial condition for the angular

With the optimal control laws uað Þt of (51), ubð Þt of (52), and ucð Þt of (53), the author illustrates the influences of the decay rate β in (29) and the weight matrix R in (31) on the control performance. Then, in Figures 3–5, the author shows the numerical simulation results for a rigid body with each control law uað Þt , ubð Þt , and ucð Þt : In Figures 3–5, the red-solid, green-dashed, and blue-dotted lines represent the state trajectories of a rigid body with uað Þt , ubð Þt , and ucð Þt , respectively.

First, as shown in Figures 3–5, we see that uað Þt , ubð Þt , and ucð Þt guarantee the

Finally, in Figures 3 and 4, we see that the state responses of a rigid body with ubð Þt show faster convergence rates to the equilibrium point than those with ucð Þt . And, in Figure 5, we see that the control efforts using ubð Þt and ucð Þt are comparable. We can explain this result by the fact that the weight matrix R whose diagonal elements are small increases the penalty on the state vector and decreases the penalty on the control input vector, which makes the system stable within a short period of time and the weight matrix R whose diagonal elements are large decreases

asymptotic stability of the equilibrium point. Second, in Figures 3 and 4, we observe that uað Þt provides more desirable control performance than ubð Þt and ucð Þt

because the design of uað Þt incorporates with the decay rate constraint.

The author designs the other by solving (27) and (28) for the TS-type fuzzy

system in (50) with α ¼ 2 and R ¼ 3 � I3�3, and the other is given by

ubðÞ¼� t Kbx tð Þ, (51)

ucðÞ¼� t Kcx tð Þ, (52)

given by

where

where

0:7999 �

209

Kc ¼

4.3 Numerical simulation results

velocity vector is <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup>rad/sec.

Kb ¼

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

We can use the result of theorem 2 to design the optimal control law for the TStype fuzzy system in (50) because the TS-type fuzzy system in (50) has the common input matrix B in (47). Then, we assume that α ¼ 2, β ¼ 0:1, and R ¼ I<sup>3</sup>�<sup>3</sup> in (28) and (29). With these values, the authors solve the LMIs in (27), (28), and (29) by using the command of "feasp" provided by the LMI Control Toolbox of MATLAB [35]. From the result, we obtain the following optimal control law for the TS-type fuzzy system in (43) that minimizes the cost function in (31):

$$u\_a(t) = -K\_a \varkappa(t),\tag{50}$$

where

$$K\_d = \begin{bmatrix} 110.0629 & -24.5318 & -47.2469 & 73.0798 & -28.6127 & -37.7649 \\ -16.3545 & 139.9846 & -60.3201 & -17.0025 & 75.0011 & -39.4506 \\ -23.6235 & -45.2400 & 185.6703 & -5.6923 & -18.6642 & 92.8441 \end{bmatrix}.$$

Since we can analyze the pure effect of the weight matrix R on the control performance without any constraint, the author designs two other optimal control laws for the TS-type fuzzy system in (50) without considering the decay rate constraint, which is given by (29), to clearly analyze the influence of the weight matrix R in (31) on the control performance. The author designs one by solving (27) Optimal Control of Fuzzy Systems with Application to Rigid Body Attitude Control DOI: http://dx.doi.org/10.5772/intechopen.82181

and (28) for the TS-type fuzzy system in (50) with α ¼ 2 and R ¼ I3�3, and one is given by

$$
\mu\_b(t) = -K\_b \mathbf{x}(t),
\tag{51}
$$

where

$$K\_b = \begin{bmatrix} 28.0708 & 2.5099 & -1.9744 & 3.5899 & 0.3747 & -0.1420 \\ & 1.6733 & 33.7936 & -4.9727 & 1.3456 & 1.7520 & 0.3518 \\ & -0.9872 & -3.7295 & 42.8319 & 1.4713 & 1.3202 & -0.2517 \end{bmatrix}.$$

The author designs the other by solving (27) and (28) for the TS-type fuzzy system in (50) with α ¼ 2 and R ¼ 3 � I3�3, and the other is given by

$$
\mu\_{\mathfrak{c}}(\mathfrak{t}) = -K\_{\mathfrak{c}}\mathfrak{x}(\mathfrak{t}),\tag{52}
$$

where

hið Þ x tð Þ ≜

fuzzy system:

Figure 2.

Aerospace Engineering

where

2 6 4

Ka ¼

208

Q<sup>6</sup>

∑<sup>9</sup> i¼1 Q<sup>6</sup>

x t \_ðÞ¼ ∑ 9 i¼1

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

Fuzzy sets Mij, i ¼ 1, ⋯, 9, j ¼ 4, ⋯, 6 used to design TS-type fuzzy model in (49) for system in (45).

the TS-type fuzzy model in (49) can be transformed into the following TS-type

We can use the result of theorem 2 to design the optimal control law for the TStype fuzzy system in (50) because the TS-type fuzzy system in (50) has the common input matrix B in (47). Then, we assume that α ¼ 2, β ¼ 0:1, and R ¼ I<sup>3</sup>�<sup>3</sup> in (28) and (29). With these values, the authors solve the LMIs in (27), (28), and (29)

MATLAB [35]. From the result, we obtain the following optimal control law for the

110:0629 �24:5318 �47:2469 73:0798 �28:6127 �37:7649 �16:3545 139:9846 �60:3201 �17:0025 75:0011 �39:4506 �23:6235 �45:2400 185:6703 �5:6923 �18:6642 92:8441

Since we can analyze the pure effect of the weight matrix R on the control performance without any constraint, the author designs two other optimal control laws for the TS-type fuzzy system in (50) without considering the decay rate constraint, which is given by (29), to clearly analyze the influence of the weight matrix R in (31) on the control performance. The author designs one by solving (27)

by using the command of "feasp" provided by the LMI Control Toolbox of

TS-type fuzzy system in (43) that minimizes the cost function in (31):

<sup>j</sup>¼<sup>1</sup> Mij xjð Þ<sup>t</sup> � � , i <sup>¼</sup> <sup>1</sup>, <sup>⋯</sup>, <sup>9</sup>,

hið Þ x tð Þ Aix tðÞþ Bu tð Þ: (49)

uaðÞ¼� t Kax tð Þ, (50)

3 7 5:

$$K\_{\varepsilon} = \begin{bmatrix} 29.1798 & 18.4270 & 18.9874 & 0.6854 & 0.2322 & -1.5053 \\ 12.2846 & 40.0169 & 18.1900 & 2.3547 & -1.7662 & -1.3049 \\ 9.4937 & 13.6425 & 49.3411 & 2.4969 & 0.9770 & -4.9648 \end{bmatrix}.$$

#### 4.3 Numerical simulation results

As the numerical simulation model, the author uses the equations of rotational motion of a rigid body given by (38) and (41), where the inertia matrix is given by (48). The author assumes that the initial conditions at the initial time t<sup>0</sup> ¼ 0 sec for the Euler axis ^<sup>e</sup> and the Euler angle <sup>ϕ</sup> are ^e tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:4896 0:2030 0:<sup>8480</sup> <sup>T</sup> and ϕð Þ¼ t<sup>0</sup> 2:4648 rad, respectively, which give q tð Þ¼ <sup>0</sup> ½0:3320 0:4618 0:1915 0:7999 � <sup>T</sup>: Note that the given <sup>ϕ</sup>ð Þ <sup>t</sup><sup>0</sup> represents an almost upside-down initial orientation of a rigid body. Also, the author assumes a rest-to-rest maneuver of a rigid body, and, thus, the author assumes that the initial condition for the angular velocity vector is <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup>rad/sec.

With the optimal control laws uað Þt of (51), ubð Þt of (52), and ucð Þt of (53), the author illustrates the influences of the decay rate β in (29) and the weight matrix R in (31) on the control performance. Then, in Figures 3–5, the author shows the numerical simulation results for a rigid body with each control law uað Þt , ubð Þt , and ucð Þt : In Figures 3–5, the red-solid, green-dashed, and blue-dotted lines represent the state trajectories of a rigid body with uað Þt , ubð Þt , and ucð Þt , respectively.

First, as shown in Figures 3–5, we see that uað Þt , ubð Þt , and ucð Þt guarantee the asymptotic stability of the equilibrium point. Second, in Figures 3 and 4, we observe that uað Þt provides more desirable control performance than ubð Þt and ucð Þt because the design of uað Þt incorporates with the decay rate constraint.

Finally, in Figures 3 and 4, we see that the state responses of a rigid body with ubð Þt show faster convergence rates to the equilibrium point than those with ucð Þt . And, in Figure 5, we see that the control efforts using ubð Þt and ucð Þt are comparable. We can explain this result by the fact that the weight matrix R whose diagonal elements are small increases the penalty on the state vector and decreases the penalty on the control input vector, which makes the system stable within a short period of time and the weight matrix R whose diagonal elements are large decreases

#### Figure 3.

Angular velocity responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52), and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and q tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:3320 0:4618 0:1915 0:<sup>7999</sup> <sup>T</sup>:

the penalty on the state vector and increases the penalty on the control input vector,

Control input responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52),

and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and

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

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

In this chapter, the author presented a theory on the optimal control of nonlinear dynamic systems by utilizing the dynamic programming approach and the inverse optimal approach. Specifically, the author employed the dynamic programming approach to derive the Hamilton-Jacobi-Bellman (H-J-B) equation associated with the optimal control problem for nonlinear dynamic systems and utilized the inverse optimal approach to avoid the task of solving the H-J-B equation numerically. Then, the author established an optimal control design for TS-type fuzzy systems to achieve the global asymptotic stability of an equilibrium point, the optimality with respect to a cost function, and the good convergence rates of state trajectories to an equilibrium point. Based on this optimal control design, the author presented a systematic way for designing the optimal control law for TS-type fuzzy

The author showed the usefulness of the optimal control design by considering the three-axis attitude stabilization problem of a rigid body. The optimal three-axis attitude stabilizing control law for a rigid body was designed, and its control performance was analyzed by numerical simulations. The numerical simulation results demonstrated that the optimal three-axis attitude stabilizing control law designed in this chapter provides desirable optimal control performance together with good

which makes the system stable within a long period of time.

convergence rates of state trajectories to an equilibrium point.

5. Conclusion

q tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:3320 0:4618 0:1915 0:<sup>7999</sup> <sup>T</sup>:

Figure 5.

systems.

211

#### Figure 4.

Quaternion responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52), and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and q tð Þ¼ <sup>0</sup> <sup>½</sup>0:<sup>3320</sup> 0:4618 0:1915 0:7999 � T:

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

Figure 5.

Control input responses of the rigid body given by (38) and (41) with control laws uað Þt of (51), ubð Þt of (52), and ucð Þ<sup>t</sup> of (53) designed in this chapter at initial conditions <sup>ω</sup>ð Þ¼ <sup>t</sup><sup>0</sup> ½ � <sup>000</sup> <sup>T</sup> rad/sec and q tð Þ¼ <sup>0</sup> ½ � <sup>0</sup>:3320 0:4618 0:1915 0:<sup>7999</sup> <sup>T</sup>:

the penalty on the state vector and increases the penalty on the control input vector, which makes the system stable within a long period of time.
