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

with a design-model obtained by conventional linear approximation techniques, the TS-type fuzzy model can be seen as a good design-model for approximating a nonlinear dynamic system because it retains the essential features of a nonlinear dynamic system with a linguistic description in terms of fuzzy IF-THEN rules, by which the TS-type fuzzy system is valid over a range of operating points within fuzzy sets. Thus, when we design the optimal controller for a nonlinear dynamic system, it is adequate to approximate a nonlinear dynamic system with a TS-type fuzzy model and to use a TS-type fuzzy model rather than a linearized designmodel. In addition, one can expect that the optimal controller designed for a TS-type fuzzy system has a wider range of optimality than the optimal controller designed for a linearized design-model because the former guarantees the optimality over a range of operating points within fuzzy sets.

As a control design example in this chapter, the three-axis attitude stabilization problem of a rigid body is considered to illustrate the optimal control design method for TS-type fuzzy systems presented in this chapter. The attitude motion of a rigid body is basically represented by a set of two Equations [36]: (i) Euler's dynamic equation, which describes the time derivative of the angular velocity vector and (ii) the kinematic equation, which relates the time derivatives of the orientation angles to the angular velocity vector. For representing the orientation angles of a rigid body, there exist several kinematic parameterizations such as Euler angles, Gibbs vector, Cayley-Rodrigues parameters, and modified Rodrigues parameters [37, 38], which are singular three-dimensional parameter representations, and quaternion (also called Euler parameters), which is a nonsingular four-dimensional parameter representation. Note that three-dimensional parameter representations exhibit singular orientations because the Jacobian matrix is singular for some orientations. On the other hand, the quaternion consists of four parameters subject to the unit length constraint and is a globally nonsingular parameter for describing the body orientation [36].

In this chapter, the equations of motion of a rigid body including dynamics and kinematics are considered. The kinematic equation of a rigid body considered in this chapter is described by the quaternion. The equations of motion of a rigid body considered in this chapter describe a system in cascade interconnection, and the backstepping method of [39] can be efficiently utilized to apply the optimal control design method presented in this chapter to the three-axis attitude stabilization problem of a rigid body.

The optimal attitude stabilization problem of a rigid body has been addressed by several researchers [40–42]. Also, there have been many studies which consider performance indices such as time and/or fuel in the formulation of the optimal attitude stabilization problem of a rigid body [43–49], in which the optimal regulation problems for angular velocity subsystem of a rigid body and for some quadratic performance indexes have mainly been addressed.

The optimal attitude control problem of the complete attitude motion of a rigid body, which includes dynamics as well as kinematics, has been investigated by many researchers: Carrington and Junkins [50] used a polynomial expansion approach to approximate the solution of H-J-B equation. Rotea et al. [51] showed that Lyapunov functions including a logarithmic term in the kinematic parameters result in linear controllers with a finite quadratic performance index. For the general quadratic performance index, they also presented sufficient conditions which guarantee the existence of a linear and suboptimal stabilizing controller. Tsiotras [52] derived a new class of globally asymptotically stabilizing feedback control laws as well as a family of exponentially stabilizing optimal control laws for the complete attitude motion of a nonsymmetric rigid body. Later, Tsiotras [53] presented a

subject to a multistage fuzzy system. Esfahani and Sichani [28] studied the problem

Since TS-type fuzzy systems essentially have a nonlinear nature due to the IF-THEN fuzzy implication and fuzzy inference, we see that a nonlinear optimal control method is suitable for designing an optimal control law for TS-type fuzzy systems. In addition, when we design an optimal control law for TS-type fuzzy systems, it is often required that the control design should allow us to control the convergence rates of state trajectories to an equilibrium point. The decay rate of the closed-loop dynamics may be used to achieve this requirement in the control design. These observations motivate the author to study an optimal control of TS-type fuzzy systems that can provide good convergence rates of state trajectories

More specifically, in this chapter, the author presents a theoretical result on the optimal control of nonlinear dynamic systems. In this theoretical result, the author presents the optimal control problem for nonlinear dynamic systems and solves this problem by utilizing the dynamic programming approach [29] and the inverse optimal approach [30]. Note that Kalman [31] first proposed the inverse optimal approach to establish some gain and phase margins of a linear quadratic regulator. Also, note that the conventional direct optimal approach is based on seeking a stabilizing controller that minimizes a given performance index. On the other hand, the inverse optimal approach avoids the task of solving the Hamilton-Jacobi-Bellman (H-J-B) equation numerically but finds a stabilizing controller first and then shows its optimality with respect to a posteriorly determined performance index. In this chapter, the author employs the dynamic programming approach to derive the H-J-B equation associated with the optimal control problem for nonlinear dynamic systems. Then, the author presents an analytic way to solve the H-J-B equation with the help of the inverse optimal approach, by which the author establishes a systematic approach for designing the optimal controller for nonlinear dynamic systems. The resulting optimal controller takes the form of state feedback

Then, based on the theoretical result presented in this chapter, the author establishes an optimal control design for TS-type fuzzy systems that guarantees the global asymptotic stability of an equilibrium point and the optimality with respect to a cost function, which incorporates a penalty on the state and control input vectors, and provides good convergence rates of state trajectories to an equilibrium point. The problem appearing in this optimal control design for TS-type fuzzy systems is given as a linear matrix inequality (LMI)-based problem. From the results, the optimal controller can be found by a simple controller design procedure, which is essentially given as LMIs. The control design involving LMIs is particularly useful in practice because LMIs can be efficiently solved by recently developed interior-point methods (e.g., [32, 33]). One of the algorithms belonging to the interior-point methods can be found in [34], and an implementation of the algorithm in [34] is included in the LMI Control Toolbox of MATLAB [35], which will be used as the solver for the LMI-

Note that the optimal controller for a nonlinear dynamic system is in general designed for a linearized design-model, which is obtained by conventional linear approximation techniques, because it is sometimes difficult to solve the nonlinear optimal control problem associated with a nonlinear dynamic system. Clearly, the optimal controller designed for a linearized design-model guarantees its optimality only at an equilibrium point used to design a linearized design-model. Compared

of optimal fuzzy H∞-tracking control design for nonlinear systems that are represented by using the TS fuzzy modeling scheme. Through these studies, the

optimal control for fuzzy systems has quite been progressed.

to an equilibrium point in this chapter.

Aerospace Engineering

LgV controller and has a relaxed control gain structure.

based problem appearing in the optimal control design.

194

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 stabilization of a rigid body with external disturbances.

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 continu-

� <sup>¼</sup> LfVxt ð Þ� ð Þ <sup>α</sup> LgVxt ð Þ ð Þ � �R�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup> , 0 (2)

is the optimal, globally asymptotically stabilizing control law for the system in

lxt ð Þ¼� ð Þ <sup>4</sup>α<sup>2</sup> LfVxt ð Þ� ð Þ <sup>α</sup> LgVxt ð Þ ð Þ � �R�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup> � �

ð Þ <sup>α</sup> � <sup>1</sup> LgVxt ð Þ ð Þ � �R�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup>

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

½ � LgVxt ð Þ ð Þ

, 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

<sup>u</sup><sup>∗</sup>ðÞ¼� <sup>t</sup> <sup>2</sup>αR�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup> (3)

lxt ð Þþ ð Þ u tð ÞTRu tð Þ h idt, (4)

þ 1 2

� � � <sup>α</sup> LgVxt ð Þ ð Þ � �R�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup>

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

(5)

(6)

(7)

ous, first, partial derivatives with respect to x tð Þ and the feedback control

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

definite matrix, achieves global asymptotic stability of the equilibrium point

u tðÞ¼�αR�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup>

V xt \_ ð Þ ð Þ u tð Þ¼�αR�<sup>1</sup>

� � �

x tðÞ¼ 0 for the system in (1) such that:

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

½ � LgVxt ð Þ ð Þ <sup>T</sup>

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

P ¼ ∞ð

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

Proof: First, the following condition holds by (2):

<sup>¼</sup> V xt \_ ð Þ ð Þ u tð Þ¼�αR�<sup>1</sup>

� � � 2

� �

, � <sup>α</sup> LgVxt ð Þ ð Þ � �R�<sup>1</sup> LgVxt ð Þ ð Þ � �<sup>T</sup> , <sup>0</sup>

a globally asymptotically stabilizing control law for the system in (1) by the

associated with the optimal control problem for the system in (1):

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

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

lxt ð Þþ ð Þ u tð ÞTRu tð Þþ LfWxt ð Þþ ð Þ LgWxt ð Þ ð Þ � �u tð Þ � � � � <sup>¼</sup> <sup>0</sup>, Wð Þ¼ <sup>0</sup> <sup>0</sup>:

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

. 0

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

0

(1) that minimizes the cost function

where lxt ð Þ ð Þ is given by

for all x tð Þ 6¼ 0 and α≥1.

Lyapunov's stability theorem [4].

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

min u tð Þ

197

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

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