2. Nonlinear optimal control

Consider the nonlinear dynamic system given by

$$
\dot{\boldsymbol{x}}(t) = \boldsymbol{f}(\boldsymbol{x}(t)) + \mathbf{g}(\boldsymbol{x}(t))\boldsymbol{u}(t), \tag{1}
$$

where <sup>f</sup> : Rn ! <sup>R</sup><sup>n</sup> and <sup>g</sup> : <sup>R</sup><sup>n</sup> ! Rn�<sup>p</sup> are smooth, vector- and matrix-valued functions, respectively, and <sup>f</sup>ð Þ¼ <sup>0</sup> <sup>0</sup>: Moreover, x tð Þ<sup>∈</sup> Rn and u tð Þ∈R<sup>p</sup> are the state and control input vectors, respectively. Throughout this chapter, we use the definitions of

$$L\_f V(\mathfrak{x}(t)) \triangleq \left(\frac{\partial V(\mathfrak{x}(t))}{\partial \mathfrak{x}(t)}\right) f(\mathfrak{x}(t)) \text{ and}$$

$$L\_{\rm g}V(\varkappa(t)) \triangleq \left(\frac{\partial V(\varkappa(t))}{\partial \varkappa(t)}\right) \mathbf{g}(\varkappa(t))\text{, where}$$

<sup>V</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup> is a scalar function [4].

In general, we can find the optimal control law for the nonlinear dynamic system in (1) by numerically solving the corresponding Hamilton-Jacobi-Bellman (H-J-B) equation. However, this is a difficult task, and, thus, we may need a simple and efficient method to find optimal control law for the nonlinear dynamic system in (1). In the following, the author presents a theory to provide the optimal control law for the nonlinear dynamic system in (1) by circumventing the task of numerically solving the H-J-B equation.

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

Proposition 1 [62]: For the nonlinear dynamic system in (1), suppose that there exists a radially unbounded and positive definite function Vxt ð Þ ð Þ that has continuous, first, partial derivatives with respect to x tð Þ and the feedback control u tðÞ¼�αR�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup> , where <sup>α</sup> <sup>≥</sup>1 is a constant and <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . 0 is a positive definite matrix, achieves global asymptotic stability of the equilibrium point x tðÞ¼ 0 for the system in (1) such that:

$$\dot{V}(\mathbf{x}(t))\Big|\_{\mathbf{u}(t)=-aR^{-1}\left[L\_{\mathbf{f}}V(\mathbf{x}(t))\right]^{T}} = L\_{\mathbf{f}}V(\mathbf{x}(t)) - a\left[L\_{\mathbf{f}}V(\mathbf{x}(t))\right]R^{-1}\left[L\_{\mathbf{g}}V(\mathbf{x}(t))\right]^{T} < \mathbf{0} \tag{2}$$

for all x tð Þ 6¼ 0: Then, the control law

$$u^\*(t) = -2aR^{-1} \left[ L\_\sharp V(\varkappa(t)) \right]^T \tag{3}$$

is the optimal, globally asymptotically stabilizing control law for the system in (1) that minimizes the cost function

$$P = \int\_0^\infty \left[ l(\varkappa(t)) + \mathfrak{u}(t)^T R \mathfrak{u}(t) \right] dt,\tag{4}$$

where lxt ð Þ ð Þ is given by

partial solution to the optimal regulation problem of a spinning rigid body by using the natural decomposition of the complete attitude motion into its kinematics and dynamics systems and the inherent passivity properties of these two systems. Bharadwaj et al. [54] presented a couple of new globally stabilizing attitude control laws based on minimal and exponential coordinates. Park and Tahk [55] have considered the problem of three-axis robust attitude stabilization of a rigid body with inertia uncertainties, and they have presented a class of new robust attitude control laws having relaxed feedback gain structures. Later, Park and Tahk [56] have extended their robust attitude control scheme of [55] to the optimal attitude control scheme by using the Hamilton-Jacobi theory of [57]. Also, Park et al. [58] have first addressed a game-theoretic approach to robust and optimal attitude

Note that, in the case of robot arm control, since the arms or hand fingers can be viewed as actuators which maneuver the attitude of the held object, the results on the attitude control of a rigid body can be applied to the attitude control of a rigid payload held by the robot arm [59]. With this relation, there have been many studies concerning the attitude control problem of a rigid body, and some remark-

The rest of this chapter is composed as follows. In Section 2, the author presents a theoretical result on the optimal control of nonlinear dynamic systems. In Section 3, the author introduces TS-type fuzzy systems and presents an optimal control design for TS-type fuzzy systems. In Section 4, the author considers the three-axis attitude stabilization problem of a rigid body as a control design example and illustrates the effectiveness of the optimal control design for TS-type fuzzy systems.

where <sup>f</sup> : Rn ! <sup>R</sup><sup>n</sup> and <sup>g</sup> : <sup>R</sup><sup>n</sup> ! Rn�<sup>p</sup> are smooth, vector- and matrix-valued functions, respectively, and <sup>f</sup>ð Þ¼ <sup>0</sup> <sup>0</sup>: Moreover, x tð Þ<sup>∈</sup> Rn and u tð Þ∈R<sup>p</sup> are the state and control input vectors, respectively. Throughout this chapter, we use the definitions of

> <sup>∂</sup>x tð Þ

<sup>∂</sup>x tð Þ 

In general, we can find the optimal control law for the nonlinear dynamic system in (1) by numerically solving the corresponding Hamilton-Jacobi-Bellman (H-J-B) equation. However, this is a difficult task, and, thus, we may need a simple and efficient method to find optimal control law for the nonlinear dynamic system in (1). In the following, the author presents a theory to provide the optimal control law for the nonlinear dynamic system in (1) by circumventing the task of numerically

x t \_ðÞ¼ f xt ð Þþ ð Þ gxt ð Þ ð Þ u tð Þ, (1)

f xt ð Þ ð Þ and

gxt ð Þ ð Þ , where

In Section 5, the author concludes this chapter with concluding remarks.

LfVxt ð Þ ð Þ <sup>≜</sup> <sup>∂</sup>Vxt ð Þ ð Þ

LgVxt ð Þ ð Þ <sup>≜</sup> <sup>∂</sup>Vxt ð Þ ð Þ

stabilization of a rigid body with external disturbances.

Consider the nonlinear dynamic system given by

able studies can be referred to [60, 61].

Aerospace Engineering

2. Nonlinear optimal control

<sup>V</sup> : <sup>R</sup><sup>n</sup> ! <sup>R</sup> is a scalar function [4].

solving the H-J-B equation.

196

$$\begin{split} l(\mathbf{x}(t)) &= -4a^2 \Big( L\_{\mathcal{f}}V(\mathbf{x}(t)) - a \Big[ L\_{\mathcal{\boldsymbol{\mathcal{E}}}}V(\mathbf{x}(t)) \Big] \mathcal{R}^{-1} \Big[ L\_{\mathcal{\boldsymbol{\mathcal{E}}}}V(\mathbf{x}(t)) \Big]^T \Big) \\ &+ 4a^2 (a - \mathbf{1}) \Big[ L\_{\mathcal{\boldsymbol{\mathcal{E}}}}V(\mathbf{x}(t)) \Big] \mathcal{R}^{-1} \Big[ L\_{\mathcal{\boldsymbol{\mathcal{E}}}}V(\mathbf{x}(t)) \Big]^T \\ &> \mathbf{0} \end{split} \tag{5}$$

for all x tð Þ 6¼ 0 and α≥1. Proof: First, the following condition holds by (2):

$$\dot{V}(\mathbf{x}(t))|\_{\mathbf{u}(t)=\mathbf{u}^\*(t)} = \left(L\_{\mathcal{f}}V(\mathbf{x}(t)) + \frac{1}{2}\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]\mathbf{u}^\*(t)\right) + \frac{1}{2}\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]\mathbf{u}^\*(t)$$

$$= \left(\dot{V}(\mathbf{x}(t))\Big|\_{\mathbf{u}(t)=-a\mathbb{R}^{-1}\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]}\right) - a\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]\mathbf{R}^{-1}\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]^T$$

$$< -a\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]\mathbf{R}^{-1}\left[L\_{\mathcal{E}}V(\mathbf{x}(t))\right]^T<\mathbf{0} \tag{6}$$

for all x tð Þ 6¼ 0 and α≥1: Since Vxt ð Þ ð Þ is a radially unbounded and positive definite function, the condition in (6) guarantees that the control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) is a globally asymptotically stabilizing control law for the system in (1) by the Lyapunov's stability theorem [4].

Next, define Wxt ð Þ ð Þ <sup>≜</sup>4α<sup>2</sup>Vxt ð Þ ð Þ and consider the following H-J-B equation associated with the optimal control problem for the system in (1):

$$\min\_{\mathbf{u}(t)} \left( l(\mathbf{x}(t)) + \mathbf{u}(t)^T R \mathbf{u}(t) + \left\{ L\_f W(\mathbf{x}(t)) + \left[ L\_\mathbf{g} W(\mathbf{x}(t)) \right] \mathbf{u}(t) \right\} \right) = \mathbf{0}, \quad W(\mathbf{0}) = \mathbf{0}. \tag{7}$$

Substituting <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) and lxt ð Þ ð Þ in (5) into the H-J-B equation in (7) yields

$$\begin{split} &\min\_{u(t) = u^\*(t)} \left( l(\mathbf{x}(t)) + u(t)^T R u(t) + \left[ L\_f W(\mathbf{x}(t)) + \left[ L\_g W(\mathbf{x}(t)) \right] u(t) \right) \right) \\ &= \min\_{u(t) = u^\*(t)} \left( \left\{ -L\_f W(\mathbf{x}(t)) + \frac{1}{4\alpha} \left[ L\_g W(\mathbf{x}(t)) \right] R^{-1} \left[ L\_g W(\mathbf{x}(t)) \right]^T \right. \\ &\left. + \frac{(a-1)}{4\alpha^2} \left[ L\_g W(\mathbf{x}(t)) \right] R^{-1} \left[ L\_g W(\mathbf{x}(t)) \right]^T \right\} + \frac{1}{4\alpha^2} \left[ L\_g W(\mathbf{x}(t)) \right] R^{-1} \left[ L\_g W(\mathbf{x}(t)) \right]^T \right. \\ &\left. + \left\{ L\_f W(\mathbf{x}(t)) - \frac{1}{2\alpha} \left[ L\_g W(\mathbf{x}(t)) \right] R^{-1} \left[ L\_g W(\mathbf{x}(t)) \right]^T \right\} \right) = 0, \quad W(\mathbf{0}) = \mathbf{0}, \end{split} \tag{8}$$

Proposition 2 [32]: If there exist a positive definite function Vxt ð Þ ð Þ and a

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

for all trajectories of a system, then the decay rate of a system is at least β: Proof: If there exist a positive definite function Vxt ð Þ ð Þ and a constant β . 0 such

that V xt \_ ð Þ ð Þ , � <sup>2</sup>βVxt ð Þ ð Þ for all trajectories of a system, then we obtain

<sup>2</sup> e�β<sup>t</sup>

� � � 2 2 <sup>λ</sup>min <sup>X</sup>�<sup>1</sup> � � <sup>≤</sup>

where <sup>X</sup> <sup>¼</sup> <sup>X</sup><sup>T</sup> . 0 is a positive definite matrix, Vxt ð Þ ð Þ , V xð Þ ð Þ <sup>0</sup> <sup>e</sup>�2β<sup>t</sup> can be

xð Þ 0

� � � 2 2

X�<sup>1</sup> 2 � � �

where λmin X�<sup>1</sup> � � and λmax X�<sup>1</sup> � � denote the minimum and maximum eigenvalues

The TS-type fuzzy model and the TS-type fuzzy control law for a system are

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,

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

given by the following IF-THEN fuzzy implications, respectively [15]:

� � � 2 2 <sup>λ</sup>min <sup>X</sup>�<sup>1</sup> � � <sup>e</sup>

, and therefore the decay rate of the system is at least β: This completes

�βt <sup>x</sup>ð Þ <sup>0</sup> � � �

x t \_ðÞ¼ Aix tðÞþ Biu tð Þ, i ¼ 1, ⋯, r: (10)

u tðÞ¼�Kix tð Þ, i ¼ 1, ⋯, r: (11)

�2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λmax X�<sup>1</sup> � �=λmin X�<sup>1</sup> q� � � �

V xt \_ ð Þ ð Þ , � <sup>2</sup>βVxt ð Þ ð Þ (9)

: Then, by Rayleigh-Ritz theorem [63],

<sup>2</sup> <sup>¼</sup> <sup>λ</sup>max <sup>X</sup>�<sup>1</sup> � �

<sup>λ</sup>min <sup>X</sup>�<sup>1</sup> � � k k <sup>x</sup>ð Þ <sup>0</sup> <sup>2</sup>

x tð Þ,

2e �2βt ,

: With a positive definite function Vxt ð Þ ð Þ <sup>≜</sup>x tð ÞTX�<sup>1</sup>

constant β . 0 such that

Vxt ð Þ ð Þ , V xð Þ ð Þ <sup>0</sup> <sup>e</sup>�2β<sup>t</sup>

represented by X�<sup>1</sup>

k k x tð Þ <sup>2</sup> <sup>2</sup> ≤

of X�<sup>1</sup>

k k <sup>x</sup>ð Þ <sup>0</sup> <sup>2</sup>e�β<sup>t</sup>

3.1 Fuzzy systems

• Plant rule i for a system:

• Control law rule i for a system:

Ai ∈ Rn�n, Bi ∈R<sup>n</sup>�p, and Ki ∈ Rp�<sup>n</sup>:

system as follows, respectively [9]:

199

the proof.

<sup>2</sup>x tð Þ

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

� � � 2 2

X�<sup>1</sup> <sup>2</sup> e�β<sup>t</sup> xð Þ 0

,respectively. Thus, we obtain k k x tð Þ <sup>2</sup> ,

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

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

� � �

3. Optimal control of fuzzy systems

, X�<sup>1</sup>

� � �

� � �

� � � 2 2 <sup>λ</sup>min <sup>X</sup>�<sup>1</sup> � � ,

we can derive the following:

X�<sup>1</sup> <sup>2</sup>x tð Þ

� � �

which implies that <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) and in (5) are solutions of the H-J-B equation in (7). In addition, by (2) and the property of <sup>R</sup> <sup>¼</sup> <sup>R</sup><sup>T</sup> . <sup>0</sup>,lxt ð Þ ð Þ in (5) satisfies lxt ð Þ ð Þ . 0 for all x tð Þ 6¼ 0 and α≥ 1: This completes the proof.

In proposition 1, we see that the globally asymptotically stabilizing control law in (3) for the nonlinear dynamic system in (1) can be found without the task of numerically solving the H-J-B equation in (7) and the control law in (3) is optimal with respect to the cost function in (4). The key point of this work is that we posteriorly determine the penalty on the state vector, which is lxt ð Þ ð Þ , rather than we priorly choose it. Sepulcher, Janković, and Kokotović [30] proposed this approach which is referred to as the inverse optimal approach.

It is remarkable that as shown in (4, 5), we can adjust the penalty on the control input vector, which is R, and the penalty on the state vector, which is lxt ð Þ ð Þ , with the weight matrix R:Indeed, we can decrease the penalty on the control input vector and increase the penalty on the state vector with a weight matrix R having small values. In this condition, we can obtain a cheap optimal control law requiring a large control effort, and this cheap optimal control law makes the nonlinear dynamic system in (1) stable within a short period of time. Note that the term "cheap" refers to the fact that the control effort is viewed as being cheap. On the other hand, we can increase the penalty on the control input vector and decrease the penalty on the state vector with a weight matrix R having large values. In this condition, we can obtain an expensive optimal control law requiring a small control effort, and this expensive optimal control law makes the nonlinear dynamic system in (1) stable within a long period of time. Note that the term "expensive" refers to the fact that the control effort is viewed as being expensive.

As shown in (3), since the constant <sup>α</sup> of the optimal control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) plays the role of a feedback gain for <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> even though the weight matrix <sup>R</sup> is predetermined to impose the penalties on the control input and state vectors, it is also remarkable that the optimal control law <sup>u</sup><sup>∗</sup>ð Þ<sup>t</sup> in (3) has a relaxed feedback gain structure.

Now, if we consider a practical application, we know that any control law for dynamic systems must provide good convergence rates of state trajectories to an equilibrium point. For achieving this requirement in the design of a control law, the decay rate can be used as a design factor to dominate convergence rates of state trajectories to an equilibrium point. Note that the decay rate of a system is defined to be the largest constant <sup>γ</sup> . 0 such that lim<sup>t</sup>!<sup>∞</sup> <sup>e</sup><sup>γ</sup><sup>t</sup> k k x tð Þ <sup>2</sup> ¼ 0 holds for all trajectories of a system, where k k x tð Þ <sup>2</sup> denotes the Euclidean norm of x tð Þ: From the definition of decay rate, the convergence rate of the system trajectory to the equilibrium point can be controlled. Also, note that stability of dynamic systems corresponds to a positive decay rate. In the following, the author presents a theory about the decay rate.

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

Proposition 2 [32]: If there exist a positive definite function Vxt ð Þ ð Þ and a constant β . 0 such that

$$
\dot{V}(\mathbf{x}(t)) \le -2\beta V(\mathbf{x}(t)) \tag{9}
$$

for all trajectories of a system, then the decay rate of a system is at least β:

Proof: If there exist a positive definite function Vxt ð Þ ð Þ and a constant β . 0 such that V xt \_ ð Þ ð Þ , � <sup>2</sup>βVxt ð Þ ð Þ for all trajectories of a system, then we obtain Vxt ð Þ ð Þ , V xð Þ ð Þ <sup>0</sup> <sup>e</sup>�2β<sup>t</sup> : With a positive definite function Vxt ð Þ ð Þ <sup>≜</sup>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, Vxt ð Þ ð Þ , V xð Þ ð Þ <sup>0</sup> <sup>e</sup>�2β<sup>t</sup> can be represented by X�<sup>1</sup> <sup>2</sup>x tð Þ � � � � � � 2 2 , X�<sup>1</sup> <sup>2</sup> e�β<sup>t</sup> xð Þ 0 � � � � � � 2 2 : Then, by Rayleigh-Ritz theorem [63], we can derive the following:

$$\|\mathbf{x}(t)\|\_{2}^{2} \leq \frac{\left\|X^{-\frac{1}{2}}\mathbf{x}(t)\right\|\_{2}^{2}}{\lambda\_{\text{min}}\left(X^{-1}\right)} < \frac{\left\|X^{-\frac{1}{2}}e^{-\beta t}\mathbf{x}(0)\right\|\_{2}^{2}}{\lambda\_{\text{min}}\left(X^{-1}\right)} \leq \frac{\left\|X^{-\frac{1}{2}}\right\|\_{2}^{2}}{\lambda\_{\text{min}}\left(X^{-1}\right)} \left\|e^{-\beta t}\mathbf{x}(0)\right\|\_{2}^{2} = \frac{\lambda\_{\text{max}}\left(X^{-1}\right)}{\lambda\_{\text{min}}\left(X^{-1}\right)} \left\|\mathbf{x}(0)\right\|\_{2}^{2}e^{-2\beta t},$$

where λmin X�<sup>1</sup> � � and λmax X�<sup>1</sup> � � denote the minimum and maximum eigenvalues of X�<sup>1</sup> ,respectively. Thus, we obtain k k x tð Þ <sup>2</sup> , ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λmax X�<sup>1</sup> � �=λmin X�<sup>1</sup> q� � � � k k <sup>x</sup>ð Þ <sup>0</sup> <sup>2</sup>e�β<sup>t</sup> , and therefore the decay rate of the system is at least β: This completes the proof.
