H∞ sampled-data fuzzy control for attitude tracking of mars entry vehicles with control constraints

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H Sampled-Data Fuzzy Control for Attitude Tracking of Mars Entry Vehicles With Control Constraints Huai-Ning Wu, Zi-Peng Wang, Lei Guo PII: DOI: Reference:

S0020-0255(16)32161-2 https://doi.org/10.1016/j.ins.2018.09.044 INS 13955

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22 December 2016 19 September 2018 20 September 2018

Please cite this article as: Huai-Ning Wu, Zi-Peng Wang, Lei Guo, H Sampled-Data Fuzzy Control for Attitude Tracking of Mars Entry Vehicles With Control Constraints, Information Sciences (2018), doi: https://doi.org/10.1016/j.ins.2018.09.044

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ACCEPTED MANUSCRIPT

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H∞ Sampled-Data Fuzzy Control for Attitude Tracking of Mars Entry Vehicles With Control Constraints Huai-Ning Wua,* , Zi-Peng Wangb,* , and Lei Guoa

and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China b School of Electrical Engineering, University of Jinan, Jinan 250022, China

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a Science

Abstract

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In this paper, the problem of designing an H∞ sampled-data fuzzy controller is investigated for attitude tracking of Mars entry vehicles with control constraints. Initially, to overcome the difficulty of Takagi-Sugeno (T-S) fuzzy modeling, the original nonlinear error system is divided into a fast subsystem and a slow subsystem on the basis of two time-scale decomposition technique, where the fast subsystem describes the attitude dynamics and the slow subsystem describes the attitude kinematics. Dynamic inversion control method is subsequently employed to obtain the angular velocity command for the slow subsystem. Then, based on the angular velocity command and the T-S fuzzy model of the fast subsystem, a tracking error fuzzy system is derived for the sampleddata fuzzy control design. The existence condition of the constrained H∞ sampled-data fuzzy controllers is provided in terms of linear matrix inequalities (LMIs). The proposed controller can exponentially stabilize the original nonlinear error system with an H∞ tracking performance, provided that the timescale separation between the fast and slow subsystems is valid. Finally, simulation results illustrate the effectiveness of the proposed method.

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Keywords: Attitude tracking; Mars entry vehicles; Fuzzy control; Sampled-data control; H∞ control; Linear matrix inequality (LMI).

1. INTRODUCTION

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Since the Mariner 4 became the first successful mission to reach Mars launched by the United States in 1965, remarkable progress has been promoted by the robotic Mars exploration missions [8]. The latest successful robotic Mars landing mission–NASA’s Mars science laboratory mission, places a mobile landing vehicle called Curiosity rover onto the surface of Gale Crater on Mars to investigate the area’s past and present environments. The future human exploration of Mars will face more difficult challenges that beyond those of the past robotic missions with the requirements of higher landing accuracy and heavier payload. Among these challenges, the most tough ones are the entry, descent and landing phase [9]. An active guided entry trajectory can significantly reduce the position errors at parachute deployment than a ballistic entry trajectory [28]. The active guided entry phase takes the Mars entry vehicle through the Martian atmosphere from hypersonic speeds to supersonic speeds whereat the parachute deployment conditions are satisfied. During the Mars atmosphere entry phase, the guidance system uses the position and attitude information provided by the on-board navigation system to minimize the downrange and crossrange errors by changing the lift vector through bank angle ∗ Corresponding authors: Zi-Peng Wang and Huai-Ning Wu. E-mail addresses: [email protected] (H.-N. Wu); [email protected] (Z.-P. Wang); [email protected] (L. Guo).

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commands. Much of the work (see, e.g., [40, 44]) is done by the guidance system to determine bank angle commands, which is then fed into the control system as a reference attitude command to calculate the desired control torques that the reaction control system thrusters need to produce [27]. The Mars entry control problem is an attitude tracking problem which is generally more difficult than the attitude stabilization problem. In the past decades, the attitude tracking problem of spacecrafts has been extensively studied in [3, 10, 19, 25–27, 48]. However, the existing results mainly focus on the continuous-time controller designs. As is well known, the sampled-data control theory has attracted great attention due to the technological appeal of digital implementations (see, for instance, [6, 12] and the references therein). Until now, there has been little progress on the attitude tracking sampled-data control design for Mars entry vehicles (MEVs). On the other hand, it has been shown that fuzzy control can offer a systematic and effective framework to solve the control design of nonlinear systems [1, 4, 5, 11, 18, 31, 38, 39, 47]. In particular, the fuzzy control technique on the basis of the Takagi-Sugeno (T-S) fuzzy model has become more and more popular in the past few decades (see, e.g., [1, 11, 18, 31, 38, 39, 47]). Recently, the attitude control problem of spacecrafts has been dealt with by this fuzzy control technique in [7, 15, 17, 21, 41, 42]. Moreover, some nice results on the sampled-data fuzzy control design have been proposed for the T-S fuzzy systems [16, 20, 24, 32, 34–36, 43, 46, 49]. These results involve mainly three approaches to deal with the sampled-data fuzzy control design problem (i.e., discrete-time system method [20, 24], switched system method [32], and time-delay system method [16, 34–36, 43, 46, 49]). More recently, a sampled-data fuzzy control method is introduced in [37] to deal with the attitude tracking problem of MEVs. But, the results of [37] cannot ensure the effectiveness of timescale separation in theory and does not taken into account some important factors, such as control constraints and disturbances. This study will consider an H∞ sampled-data fuzzy control design for the attitude tracking of MEVs. Initially, two time-scale decomposition (TTSD) method is utilized to divide the original nonlinear error system into a fast subsystem describing the attitude dynamics and a slow subsystem describing the attitude kinematics. Subsequently, the dynamic inversion control (DIC) method is introduced to obtain the angular velocity command for the slow subsystem. Then, with the angular velocity command and the T-S fuzzy model of the fast subsystem, a tracking error fuzzy system with disturbances is established to design a sampled-data fuzzy controller which can provide the desired control torques. To avoid the saturation of actuators in practical applications and attenuate the effect of the disturbances on the angular velocity tracking error, an H∞ sampled-data fuzzy control design with constraints is developed on the basis of a timedependent functional. The existence condition of the constrained H∞ sampled-data fuzzy controllers is then presented in terms of linear matrix inequalities (LMIs), which is solved efficiently via the existing LMI optimization technique [2, 14]. The resulting controller can ensure that the closed-loop of the original nonlinear error system is exponentially stable and satisfies an H∞ tracking performance, provided that the timescale separation between the fast and slow subsystems is valid. Finally, simulation results are given to demonstrate the effectiveness of the developed method. The main novelty and contribution of this paper in comparison with the existing works can be summarized as follows: (i) A novel constrained H∞ sampled-data fuzzy control design is proposed for MEVs, while the existing results mainly focus on a guidance law design for MEVs (see e.g., [40, 44]) or a continuous-time controller design for rigid spacecrafts (see e.g., [3, 10, 19, 25–27, 48]). (ii) An LMI-based condition is derived for verifying the effectiveness of the timescale separation under the proposed controller, however, the existing attitude tracking control results for spacecrafts (see e.g., [15, 19, 37]) cannot ensure the effectiveness of timescale separation in theory. Notations: N, N+ are the set of nonnegative and positive integers, respectively. The set of real and nonnegative real numbers are denoted by R and R+ , respectively. n-dimensional Euclidean space with the norm k · k and the set of all n × m matrices are denoted by Rn and Rn×m , respectively. For symmetric matrix M, M > 0 (> 0, < 0, 6 0) represents that it is positive-definite (positive-semidefinite, negative-definite, negative-semidefinite, respectively). λmin (·) (λmax (·)) stands for the minimal (maximal) eigenvalues of a matrix, respectively. A diagonal block matrix is T denoted by diag{·}. 0"n (In ) stands # for " the n × #n zero (identity) matrix. M denotes the transpose of M and a symmetric A B A B matrix is defined as , . BT C ∗ C

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2. Preliminaries and problem formulation

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As the nearest neighbor of planet Earth, Mars has attracted more attention than other planets for decades. In order to obtain the first-hand scientific data of Mars topography and chemical composition, landing a vehicle on the surface of Mars and performing in situ exploration is a prerequisite. Since the 1970s humans commenced Mars landing exploration missions. So far, more than two-thirds of the Mars missions ended in failure, and only seven spacecraft successfully landed on the surface of Mars. The Mars entry phase plays a vital role in the entire Mars exploration mission, which is challenging in many different ways [27]. Therefore, it is an interesting topic to develop some advanced control strategies to the Mars entry problem. Generally speaking, the entry problem can be separated into two parts: guidance design and control design. Considering all relevant constraints, such as initial and final constraints, heat flux, aerodynamic acceleration, and process constraints on dynamic pressure, the guidance system calculates the entry trajectory (see e.g., [40, 44]). Then the control system generates attitude commands to track the trajectory generated by the guidance system considering control constraints and disturbances.

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2.1. Attitude Model of MEVs The motion equations of the MEVs in terms of kinematics and dynamics are represented as   ˙ = G(σ(t))ω(t)  σ(t)   J ω(t) ˙ = S (ω(t))Jω(t) + u(t) + d(t)

(1)

J > 0 ∈ R3×3 denotes the inertia matrix and

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where σ(t) ∈ R3 denotes the vector of the Modified Rodrigues Parameter (MRP). G(σ(t)) = 12 [I3 + σ(t)σT (t) − S (σ(t)) − I23 (1 + σT (t)σ(t))] is a nonlinear transformation matrix, ω(t) ∈ R3 represents the vector of the angular velocity, S (ω(t)) ∈ R3×3 and    0 ω3 (t) −ω2 (t)    0 ω1 (t)  S (ω(t)) =  −ω3 (t)   ω2 (t) −ω1 (t) 0

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J = diag{J1 , J2 , J3 } = diag{J xx , Jyy , Jzz }

u(t) ∈ R3 represents the control torques. d(t) ∈ R3 stands for the vector of exogenous disturbance. Fig. 1 gives a  configuration of MEV [30].

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Figure 1. The configuration of MEV.

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Remark 1. It should be pointed out that the attitude model of MEVs in (1) is obtained from [26, 27], which has three degrees of freedom. This model is based on the assumption that the MEV is a rigid body and its mass is constant. Moreover, we neglect the effect of some important factors in the model (1) such as the uncertainties (e.g., uncertainties of aerodynamic coefficients and inertias) and actuator failures, which will be further considered in future research activities.

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For the system (1), one of feedback design objectives is to ensure that the attitude vector σ(t) can track the desired reference attitude trajectory σd (t) = [σd1 (t) σd2 (t) σd3 (t)]T ∈ R3 . In this study, we assume that σd (t) is twice differentiable and satisfy kσdg k 6 α1g , |σ ˙ dg k 6 α2g

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where α1g > 0 and α2g > 0, g = 1, 2, 3 are constant values. Let ωd (t) ∈ R3 be the reference angular velocity vector, which will be viewed as the control-like input vector to be designed in this study. The attitude and angular velocity tracking errors are defined as eσ (t) , σ(t) − σd (t) and eω (t) , ω(t) − ωd (t), respectively. Then, the tracking error system is obtained from (1) as follows:   ˙ d (t)  e˙ σ (t) = G(σ(t))ω(t) − σ (2)   J e˙ (t) = S (ω(t))Jω(t) + u(t) + d(t) − J ω ˙ d (t). ω Notice that it is very difficult to build the fuzzy modeling directly from (2). To overcome this difficulty, the TTSD method is utilized to divide the original nonlinear error system (2) into a fast subsystem and a slow subsystem. The fast subsystem and the slow subsystem describe the attitude dynamics and the attitude kinematics, respectively. Assume that the state of fast subsystem is at its equilibrium condition ω = ωd when coping with the slow dynamics, and that the state of the slow subsystem remains constant when coping with the fast dynamics. The TTSD approach is based on the singular perturbation theory [23], which can be used to deal with many practical systems such as singularly perturbed systems [22, 33], parabolic PDE Systems [36], and so on.

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2.2. DIC of slow subsystem

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Substituting the equilibrium solution ω = ωd of the fast subsystem into the slow subsystem in (2), yields the following slow subsystem: e˙ σ (t) = G(σ(t))ωd (t) − σ ˙ d (t).

(3)

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Dynamic inversion is a technique for control law design in which feedback is used to linearize nonlinear system to be controlled and to provide the desired dynamic response (see [29]). Next, we apply the DIC method to linearize the slow error subsystem (3) and to provide the desired attitude tracking error dynamics. Let 0 < K1σ ∈ R3×3 and 0 < K2σ ∈ R3×3 be the designed matrices. The control-like vector ωd (t) is selected as follows: Z −1 ωd (t) = G (σ(t))[σ ˙ d (t) − K1σ eσ (t) − K2σ eσ (s)ds]. (4)

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Substituting (4) into (3), yields the following desired attitude tracking error dynamics: Z e˙ σ (t) + K1σ eσ (t) + K2σ eσ (s)ds = 0.

(5)

Therefore, the matrices K1σ and K2σ can be selected carefully to achieve the desired tracking speed for some given reference attitude trajectory. R Define x˜1 (t) , eσ (t)dt and x˜2 (t) , eσ (t). Then it follows from (5) that " h iT where x˜(t) = x˜1T (t) x˜2T (t) and A˜ =

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x˙˜(t) = A˜ x˜(t) # I3 . −K1σ 4

(6)

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Let us first choose a Lyapunov function candidate for system (6) as V s (t) = x˜T (t)P˜ x˜(t)

(7)

where P˜ > 0 ∈ R6×6 . Along the solution of system (6), we obtain ˜ x˜(t). V˙ s (t) = 2 x˜T (t)P˜ x˙˜(t) = x˜T (t)[P˜ A˜ + A˜ T P]

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Obviously, for the chosen matrices K1σ and K2σ , the slow subsystem (3) is exponentially stable if there exists a matrix P˜ > 0 satisfying P˜ A˜ + A˜ T P˜ < 0. Remark 2. Note that the tracking performance depends on the chosen matrices K1σ and K2σ . K1σ is designed to achieve quick tracking response speed and K2σ is to eliminate the steady-state tracking error. However, it may make the TTSD invalid and even cause the system to be unstable if they are chosen to be unsuitable (e.g., K1σ and K2σ are very large). 2.3. Fuzzy modeling of fast subsystem

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The fast subsystem in (2) is represented as

e˙ ω (t) = A(ω(t))ω(t) + Bu(t) + Bd(t) − ω ˙ d (t) where c1 = (J1 − J2 )J3−1 , c2 = (J2 − J3 )J1−1 , c3 = (J3 − J1 )J2−1 , B = J −1 , and   0  A(ω(t)) =  c3 ω3 (t)  0

0 c2 ω2 (t) 0 0 c1 ω1 (t) 0

(9)

    .

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Next, the T-S fuzzy model of the fast subsystem (9) is established via the local sector nonlinearity method in [31]. Take the angular velocities ωg (t), g = 1, 2, 3 as the premise variables. We suppose that ωg (t) ∈ [ω ¯ g,1 ω ¯ g,2 ], g = 1, 2, 3. Then, ωg (t) can be expressed as follows:

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ωg (t) = µg,1 (ωg (t))ω ¯ g,1 + µg,2 (ωg (t))ω ¯ g,2 2 X = µg,p (ωg (t))ω ¯ g,p

(10)

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where

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µg,p (ωg (t)) > 0, p = 1, 2,

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µg,p (ωg (t)) = 1.

(11)

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Solving the equations in (10) and (11), we can get the membership functions as follows: ωg (t) − ω ¯ g,1 ω ¯ g,2 − ω ¯ g,1 ω ¯ g,2 − ωg (t) µg,2 (ωg (t)) , . ω ¯ g,2 − ω ¯ g,1 µg,1 (ωg (t)) ,

(12)

i−1−4·[q1 (i)−1] Define q1 (i) , 1 + fix( i−1 ), q3 (i) , i − 4 · [q1 (i) − 1] − 2 · [q2 (i) − 1], where fix(·) 4 ), q2 (i) , 1 + fix( 2 stands for the fix function. Then, the fast subsystem (9) is represented as follows: Plant Rule i: IF ω1 (t) is ω ¯ 1,q1 (i) , ω2 (t) is ω ¯ 2,q2 (i) , and ω3 (t) is ω ¯ 3,q3 (i) ,

THEN e˙ ω (t) = Ai ω(t) + Bu(t) + Bd(t) − ω ˙ d (t) 5

(13)

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where ω(t) = [ω1 (t) ω2 (t) ω3 (t)]T , and    0 0 c2 ω ¯ 2,q2 (i)    ¯ 3,q3 (i) 0 0 Ai =  c3 ω  .  0 c1 ω ¯ 1,q1 (i) 0

i=1

where ϑi (ω(t)) =

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µg,qg (i) (ωg (t)). Moreover, we can verify that 8 X

ϑi (ω(t)) > 0, i = 1, 2, · · · , 8,

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ˆ ϑi (ω(t))Ai eω (t) + Bu(t) + Bd(t)

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e˙ ω (t) =

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From (14), we have

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By the singleton fuzzifier, product inference and the center average defuzzifier, the overall fuzzy system can be obtained as 8 X ϑi (ω(t))Ai ω(t) + Bu(t) + Bd(t) − ω ˙ d (t) (14) e˙ ω (t) =

(15)

(16)

h i 8 ˆ = dT (t) dT (t) T , and d0 (t) = P ϑi (ω(t))Ai ωd (t) − ω where B = [I3 B], d(t) ˙ d (t) is viewed as the design disturbance of 0 i=1

the fuzzy system (16).

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2.4. Problem formulation In this work, based on the fuzzy model (13), we consider the following sampled-data fuzzy controller: Controller Rule j: IF ω1 (t) is ω ¯ 1,q1 ( j) , ω2 (t) is ω ¯ 2,q2 ( j) , and ω3 (t) is ω ¯ 3,q3 ( j) ,

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THEN u(t) = K j eω (tk ), t ∈ [tk , tk+1 )

(17)

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where K j ∈ R3×3 , j ∈ S , {1, 2, · · · , 8} are the control gains to be designed and {tk } is a sampling time sequence satisfying {tk } ∈ S(h) , {tk , k ∈ N|t0 = 0, ς 6 tk+1 − tk 6 h, for some 0 < ς 6 h}. Then we have the following overall fuzzy controller: 8 X u(t) = ϑ j (ω(tk ))K j eω (tk ). (18) j=1

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Denote ρ(t) , t − tk , t ∈ [tk , tk+1 ). Substituting (18) into (16), the closed-loop fuzzy system is obtained as follows:

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e˙ ω (t) =

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ˆ ϑi (ω(t))ϑ j (ω(tk ))[Ai eω (t) + BK j eω (t − ρ(t))] + Bd(t).

(19)

In this study, we consider the following H∞ tracking performance for the fuzzy system (16): Z

t0

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eTω (t0 )Peω (t0 )

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(20)

t0

where t f denotes the terminal time of control and t f ∈ S(h), γ > 0 is a given attenuation level, 0 6 N1 = C NT 1 C N1 ∈ R3×3 and 0 6 N2 = DTN2 DN2 ∈ R3×3 , and P > 0 ∈ R3×3 . The definitions are introduced for system (19) as follows: 6

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Definition 1. Given a sampling time sequence satisfy {tk } ∈ S(h). The disturbance-free fuzzy system of (19) (i.e., ˆ = 0) is exponentially stable, if there exist scalars β1 > 0, β2 > 0, and β3 > 0 so that d(t) keω (t)k 6 β2 keω (t0 )ke−β3 (t−t0 ) , ∀t > t0 , keω (t0 )k 6 β1 .

(21)

Definition 2. The tracking error fuzzy system (19) is exponentially stable with γ-disturbance attenuation, if the disturbance-free fuzzy system of (19) is exponentially stable and the H∞ tracking performance in (20) holds for some ˆ prescribed level γ > 0 in the presence of d(t).

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In general, we want to find an attenuation level as small as possible to achieve a good disturbance attenuation performance. However, a very small attenuation level usually results in high control gains. Therefore, the saturation of actuators is necessary to be considered in practical control applications. To this end, the control constraints are imposed in the design as follows: |ug (t)| 6 ug,max (22)

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where ug,max , g = 1, 2, 3 are the given positive scalars. The design objective is to search an H∞ sampled-data fuzzy controller as in (18) with constraints (22) so that the tracking error fuzzy system (19) is exponentially stable with γ-disturbance attenuation. Moreover, the resulting ˆ = 0 and achieve the following controller can stabilize exponentially the original tracking error system (2) with d(t) H∞ tracking performance: Z tf (eTω (s)N1 eω (s) + uT (s)N2 u(s))ds t0

6 x˜T (t0 )P˜ x˜(t0 ) + eTω (t0 )Peω (t0 ) + γ2

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(23)

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ˆ where P˜ > 0 and P > 0 are defined in (7) and (20), respectively. The following lemma will be in the presence of d(t), useful to derive the main results. Lemma 1. [6] For given scalars a and b satisfying

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p set c = 12 |4b − a2 | and define

√ b > 0, a > −2 b

1 2c c arctanh a , 2 a, 2c 1 c arctan a , π 2c ,

√ if a > 2 b √ if a = 2 b

√ if 0 < a < 2 b √ if − 2 b < a 6 0.

(24)

(25)

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Let α, T , and δ be given positive scalars, τ ∈ (0, T ], and let φ(t, φ0 ) be the solution of the initial value problem v (φ2 + aφ + b), t > α, φ(α) = φ0 > 0. φ˙ = − τ+δ

(26)

Then there exists a positive scalar % that only depends on δ, a, b, and T such that for some initial value φ∗0 ∈ (0, %), the solution φ(t, φ∗0 ) has the following property: ˙ < 0 for t ∈ [α, α + τ]. φ(α + τ) = 0, φ(t)

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(27)

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3. Controller design In this section, an H∞ sampled-data fuzzy control design for MEVs will be presented. To this end, a timedependent functional is first constructed for the tracking error fuzzy system (19). For any {tk } ∈ S(h), we introduce the following function ψ : [t0 , ∞) → R+ : ψ(t) = tk+1 − t, t ∈ [tk , tk+1 ), k ∈ N.

(28)

φ˙ k = − has the following property:

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Denote hk , tk+1 − tk , k ∈ N. Then, it follows from Lemma 1 that for some initial value φ∗k,0 ∈ (0, %) in which % > 0 √ depends on k > 0, b > 0, a > −2 b, and hk , the unique solution φk (t, φ∗k,0 ) of the initial value problem v (φ2 + aφk + b), t > tk , φk (tk ) = φ∗k,0 hk + k k φk (tk+1 ) = 0, φ˙ k (t) < 0, t ∈ [tk , tk+1 ].

For any {tk } ∈ S(h), a function φ : [t0 , ∞) → R+ is defined as

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φ(t) = φk (t, φ∗k,0 ), t ∈ [tk , tk+1 ), k ∈ N.

(29)

(30)

Notice that hk 6 h, one can see from (29) and (30) that φ(t) has the property

− φ(tk+1 ) = 0, k ∈ N, 0 6 φ(t) 6 %, t > t0 ; v ˙ 6 − (φ2 (t) + aφ(t) + b) < 0, t , tk , k ∈ N. φ(t) h¯

(31) (32)

where h¯ = h +  and  = max {k }. Notice ψ(t) and φ(t) have some similar properties, which are nonnegative, k

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− − ψ(tk+1 ) = φ(tk+1 ) = 0, and continuously decreasing on (tk , tk+1 ). With the above functions ρ(t), ψ(t), and ρ(t), we consider the following functional for system (19): Z t T V f (t) = eω (t)Peω (t) + ψ(t) e˙ Tω (s)R˙eω (s)ds

(33)

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t−ρ(t) T +ψ(t)[˜eω (t)X1 e˜ ω (t) + 2(eω (t) − e˜ ω (t))T X2 e˜ ω (t)] +φ(t)˜eTω (t)(ψ(t)S 1 + (h¯ − ψ(t))S 2 )˜eω (t)

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where e˜ ω (t) = eω (t) − eω (t − ρ(t)), P is the same as that in (20), R > 0 ∈ R3×3 and S l > 0 ∈ R3×3 , l = 1, 2, − − X1 = X1T ∈ R3×3 , and X2 ∈ R3×3 . Considering ρ(tk ) = 0, e˜ ω (tk ) = 0, k ∈ N, and ψ(tk+1 ) = φ(tk+1 ) = 0, k ∈ N, we have V f (t0 ) = V f (t0+ ) = eTω (t0 )Peω (t0 )

+ − V f (tk+1 ) = V f (tk+1 ) = V f (tk+1 ) = eTω (tk+1 )Peω (tk+1 ), k ∈ N.

(34)

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Then, due to the continuity of ρ(t), ψ(t), φ(t), and e˜ ω (t) on (tk , tk+1 ) for all k ∈ N, it follows from (34) that V f (t) is continuous on [t0 , +∞). For brevity, we introduce the following notations: I1 = [I3 03 03 ] , I2 = [03 I3 03 ] , I3 = [03 03 I3 ] h i I4 = [I3 − I3 03 ] , Ai j = Ai + BK j − BK j − I3 .

For t ∈ [tk , tk+1 ), along the solution of the fuzzy system (19) and considering (31) and (32), we obtain Z t D+ V f (t) 6 2eTω (t)P˙eω (t) − e˙ Tω (s)R˙eω (s)ds +ψ(t)˙eTω (t)R˙eω (t)

t−ρ(t) − e˜ Tω (t)X1 e˜ ω (t)

8

− 2(eω (t) − e˜ ω (t))T X2 e˜ ω (t)

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+2ψ(t) e˜ Tω (t)X1 + (eω (t) − e˜ ω (t))T X2 e˙ ω (t) v − (φ2 + aφ + b)˜eTω (t)(ψ(t)S 1 + (h¯ − ψ(t))S 2 )˜eω (t) h¯ +2φ(t)˜eTω (t)(ψ(t)S 1 + (h¯ − ψ(t))S 2 )˙eω (t) +φ(t)˜eTω (t)(S 2 − S 1 )˜eω (t)

(35)

8 X 8 X i=1 j=1

6

8 X 8 X i=1 j=1

+ρ(t) +

Z

ϑi (ω(t))ϑ j (ω(tk ))ξT (t)Mi j [˜eω (t) −

t

t−ρ(t)

e˙ ω (s)ds]

ϑi (ω(t))ϑ j (ω(tk ))ξT (t)(Mi j I2 + IT2 MiTj )ξ(t)

8 X 8 X

ϑi (ω(t))ϑ j (ω(tk ))ξT (t)Mi j R−1 MiTj ξ(t)

i=1 j=1

t

Z

AN US

0 = 2

CR IP T

where D+ V f (t) represents the upper right Dini derivative of V f (t). Rt Let ξ(t) = [eTω (t) e˜ Tω (t) e˙ Tω (t)]T . Since e˜ ω (t) = eω (t) − eω (t − ρ(t)) = t−ρ(t) e˙ ω (s)ds, for any matrices Mi j ∈ R9×3 , i, j ∈ S, we have

t−ρ(t)

e˙ Tω (s)R˙eω (s)ds.

(36)

Letting P = [P1 P2 P3 ]T and using (19), give the following relationship: 8 X 8 X i=1 j=1

8 X 8 X i=1 j=1

ˆ ϑi (ω(t))ϑ j (ω(tk ))ξT (t)(PAi j + ATi j PT )ξ(t) + 2ξT (t)PBd(t).

(37)

ED

=

ˆ − e˙ ω (t)] ϑi (ω(t))ϑ j (ω(tk ))(Ai eω (t) + BK j eω (t − ρ(t))) + Bd(t)

M

0 = 2ξT (t)P[

Considering ρ(t) 6 h¯ − ψ(t) and using (35)–(37), we have

PT

ˆ D+ V f (t) + eTω (t)N1 eω (t) + uT (t)N2 u(t) − γ2 dˆT (t)d(t) 8 X 8 X ¯ 6 ϑi (ω(t))ϑ j (ω(tk ))ηT (t)Ξi j (t, h)η(t)

AC

where

CE

i=1 j=1

η(t) = [ξT (t) φ(t)˜eTω (t) dˆT (t)]T ¯ = Ξ1i j + ψ(t)Ξ2i j (h) ¯ + (h¯ − ψ(t))Ξ3i j (h) ¯ Ξi j (t, h)  ij  ij  Ω10 + Υ j Ω20 PB    Ξ1i j =  ∗ 0 0   2  ∗ ∗ −γ I   ij i j ¯ Ω (h) ¯ 0   Ω11 (h) 21   v ¯ =  Ξ2i j (h) ∗ − h¯ S 1 0    ∗ ∗ 0  ij  ¯ + Mi j R−1 M T Ωi j (h) ¯ 0   Ω12 (h) ij 22   ¯ =  Ξ3i j (h) ∗ − hv¯ S 2 0    ∗ ∗ 0 9

(38)

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Υ j = IT1 N1 I1 + IT4 K Tj N2 K j I4

Ωi10j = IT1 PI3 + IT3 PI1 − IT2 X1 I2 − IT4 X2 I2

−IT2 X2T I4 + Mi j I2 + IT2 MiTj + PAi j + ATi j PT

Obviously, if the following inequality holds: 8 8 X X

¯ <0 ϑi (ω(t))ϑ j (ω(tk ))Ξi j (t, h)

then we can obtain

AN US

i=1 j=1

CR IP T

¯ = IT RI3 + IT X1 I3 + IT X1 I2 + IT X2 I3 Ωi11j (h) 3 2 3 4 bv T T T +I3 X2 I4 − I2 S 1 I2 h¯ 1 bv T ij ¯ Ω12 (h) = − I2 S 2 I2 , Ωi20j = IT2 (S 2 − S 1 ) ¯h 2  av  ij ¯ Ω2l (h) = − IT2 + IT3 S l , l = 1, 2. 2h¯

ˆ 6 0. D+ V f (t) + eTω (t)N1 eω (t) + uT (t)N2 u(t) − γ2 dˆT (t)d(t)

Therefore, we have the following result:

(39)

(40)

M

Theorem 1. Consider the tracking error fuzzy system (16) and the sampled-data fuzzy controller (18) with {tk } ∈ S(h). For given scalars γ > 0, a and b satisfying (24), r1 , r2 and r3 , the closed-loop fuzzy system (19) is exponentially stable ¯ K¯ j , and with γ-disturbance attenuation, if there exist 3 × 3 matrices P¯ > 0, X¯ 1 = X¯ 1T , X¯ 2 , R¯ > 0, S¯ l > 0, l = 1, 2, W, ¯ 9 × 3 matrices Mi j , i, j ∈ S satisfying the following LMIs: ¯ li j < 0, l = 1, 2, i, j ∈ S Π

ED

where

(41)

AC

CE

PT

¯ 1i j Π

 ¯ ij ¯ ij ¯ TC T IT K¯ T DT   Φ111 Φ112 LB IT1 W N1 4 j N2    0 0   ∗ −vS¯ 1 0  =  ∗ ∗ −γ2 I 0 0    ∗ ∗ −I 0  ∗  ∗ ∗ ∗ ∗ −I  ij  ¯ ¯ i j LB h M ¯ i j IT W ¯ TC T IT K¯ T DT   Φ Φ N j N 1 4 211 212  1 2   0 0 0  ∗ −vS¯ 2 0   ∗ 2  ∗ −γ I 0 0 0 =    ∗ ∗ ∗ −hR¯ 0 0    ∗ ∗ ∗ −I 0  ∗  ∗ ∗ ∗ ∗ ∗ −I ¯ i j + hΩ ¯ ij , Φ ¯ i j + hΩ ¯ ij ¯ ij = Ω = Ω 10 11 112 20 21 ij ij ¯ ij ij ¯ ¯ ¯ ¯ = Ω + hΩ , Φ = Ω + hΩi j

¯ 2i j Π

¯ ij Φ 111 ¯ Φi j 211

10

12

212

20

22

¯ i j = IT PI ¯ 1 − IT2 X¯ 1 I2 − IT4 X¯ 2 I2 ¯ 3 + IT3 PI Ω 1 10 ¯ iTj ¯ i j I2 + IT2 M −IT2 X¯ 2T I4 + M ¯ 1 + IT1 W ¯ T ATi LT − LWI ¯ 3 +LAi WI T ¯ T T T T −I3 W L + LBK¯ j I4 + I4 K¯ j BT LT 10

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¯ i j = IT RI ¯ 3 + IT2 X¯ 1 I3 + IT3 X¯ 1T I2 Ω 3 11 bv +IT4 X¯ 2 I3 + IT3 X¯ 2T I4 − IT2 S¯ 1 I2 h bv i j ¯ = − IT S¯ 2 I2 , LT = [r1 I3 r2 I3 r3 I3 ] Ω 12 h 2 ¯ i j = (− av IT + IT )S¯ l , l = 1, 2 ¯ i j = 1 IT (S¯ 2 − S¯ 1 ), Ω Ω 3 20 2l 2 2 2h 2

¯ −1 , j ∈ S K j = K¯ j W and the matrix P in (20) is given by

¯ −T P¯ W ¯ −1 . P=W

CR IP T

in which v is defined by (25). Moreover, the control gains are given as (42) (43)

AN US

Proof: See Appendix A.  Note that Theorem 1 presents the existence condition of the H∞ sampled-data fuzzy controllers in terms of LMIs so that the fuzzy system (19) is exponentially stable with γ-disturbance attenuation. Remark 3. From the proof of Theorem 1, the LMIs (41) imply that D+ V f (t) < 0, t ∈ [tk , tk+1 ), k ∈ N. Moreover, we get from (34) that V f (tk ) > 0 and V f (tk+1 ) > 0, k ∈ N. Then, we can obtain that V f (t) > 0 for any t ∈ [tk , tk+1 ], k ∈ N. Suppose that on the contrary, there exists t¯ ∈ (tk , tk+1 ) such that V f (t¯) 6 0. Since D+ V f (t) < 0, t ∈ [tk , tk+1 ), we have − V f (t) 6 0 for any t ∈ [t¯, tk+1 ). Due to the continuity of V f (t) on [t0 , +∞), we get V f (tk+1 ) = V f (tk+1 ) < 0, which deduces a contradiction by the fact V f (tk+1 ) > 0. Therefore, V f (t) is positive definite on [t0 , +∞).

ED

M

Remark 4. It should be pointed out that it is an interesting issue that how to reduce the conservatism of the conditions in Theorem 1. Fortunately, there exist some nice results to design a sampled-data fuzzy controller in [32, 34, 35] and the references therein, which can be applied to relax the presented conditions in this paper. Inspired by the above results, we will construct a novel Lyapunov functional and consider the property of membership functions to reduce the conservatism of this paper in future research activities.

CE

PT

It is immediate from the proof of Theorem 1 that if there exist matrices P¯ > 0, X¯ 1 = X¯ 1T , X¯ 2 , R¯ > 0, S¯ l > 0, l = 1, 2, ¯ ¯ i j , i, j ∈ S satisfying the inequalities of (41), then (39) holds, and thus (40) is satisfied. Integrating (40) W, K¯ j , and M from t0 to t yields Z t V f (t) + (eTω (s)N1 eω (s) + uT (s)N2 u(s))ds t0

6 V f (t0 ) + γ

2

Z

t

ˆ dˆT (s)d(s)ds

for all t0 6 t 6 t f . R tf ˆ 6 γ−2 dˆmax , where dˆmax > 0. If the following inequality is fulfilled: Assume that the t dˆT (s)d(s)ds

AC

(44)

t0

0

eTω (t0 )Peω (t0 ) + dˆmax 6 ν

(45)

where ν is a positive scalar, then it is immediate from (44) that the solution of system (19) stays in the following ellipsoid: E , {eω (t) ∈ R3 |V f (t) 6 ν}. (46) Then, we have the following theorem:

Theorem 2. For given scalars γ > 0, a and b satisfying (24), r1 , r2 , r3 , and ε > 0. Suppose that there exist matrices ¯ K¯ j , and M ¯ i j , i, j ∈ S satisfying the LMIs (41). Then for all t0 6 t 6 t f , P¯ > 0, R¯ > 0, X¯ 1 = X¯ 1T , X¯ 2 , S¯ l > 0, l = 1, 2, W, 11

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the control constraints in (22) are enforced for the sampled-data fuzzy controller (18) with gain matrices in (42) if the following LMIs hold: " # dˆmax − ν eTω (t0 ) (47) ¯ +W ¯ T ) + ε2 P¯ 6 0 ∗ −ε(W " # −X¯ νK¯ j 6 0, j ∈ S (48) ∗ −νP¯ ¯ gg 6 u2g,max [X]

(49)

CR IP T

where X¯ = X¯ T ∈ R3×3 and [·]gg denotes the (g, g) element of a matrix. ¯ P¯ −1 W ¯ T > ε(W ¯ +W ¯ T ) − ε2 P¯ for some given ε > 0, it is immediate from (47) that Proof: Using the fact W " # dˆmax − ν eTω (t0 ) ¯ P¯ −1 W ¯ T 6 0. ∗ −W

(50)

Using Schur complement and (43), we obtain that (50) is equivalent to (45), which means that the solution of eω (t) of the fuzzy system (19) stays in the ellipsoid (46). Thus, from (34) and (46), we get

1

1

AN US

V f (tk ) = eTω (tk )Peω (tk ) 6 ν.

(51)

Letting x(tk ) = ν− 2 P 2 eω (tk ), then from (51), we have xT (tk )x(tk ) 6 1, which implies that xT (tk )[x(tk )xT (tk ) − I]x(tk ) 6 0. Thus, we get x(tk )xT (tk ) 6 I. Moreover, pre- and post-multiplying both sides of (48) with diag{I, W T } and its transpose, we have " # −X¯ νK j 6 0, j ∈ S. (52) ∗ −νP

M

Then, using Schur complement, (52) is equivalent to

ED

¯ j ∈ S. νK j P−1 K Tj 6 X,

(53)

Define ku(t)kg , |ug (t)|. From (18), we have that for any t0 6 t 6 t f

CE

PT

max ku(t)kg = max 6 max = max 6 max

AC

6 max

q [u(t)uT (t)]gg q [K j eω (tk )eTω (tk )K Tj ]gg q 1 1 [νK j P− 2 x(tk )xT (tk )P− 2 K Tj ]gg q [νK j P−1 K Tj ]gg q ¯ gg . [X]

(54)

Therefore, from (49) and (54), we obtain (22). This completes the proof.  Define ρ¯ , γ2 . Then, based on Theorems 1 and 2, for given scalars a, b, r1 , r2 , r3 , and ε, a suboptimal constrained H∞ sampled-data fuzzy control design for the fast subsystem (19) is formulated as the following constrained optimization problem: min ρ¯ subject to LMIs (41) and (47)–(49) U

(55)

¯ K¯ j , M ¯ i j , i, j ∈ S} is a set of decision where U , {ρ¯ > 0, P¯ > 0, X¯ = X¯ T , X¯ 1 = X¯ 1T , X¯ 2 , R¯ > 0, S¯ l > 0, l = 1, 2, W, variables. 12

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Remark 5. Observe that for given a, b, r1 , r2 , r3 , and ε, the optimization problem (55) is solved by utilizing the Matlab’s LMI toolbox [14]. The problem is then how to search the optimal parameters of a, b, r1 , r2 , r3 , and ε in order to minimize ρ. ¯ As in [13, 45], one can first seek a set of initial scaling parameters such that LMIs are feasible. Then, applying the program fminsearch in Matlab, one can obtain a locally convergent solution to (55). Moreover, we can obtain that the number of decision matrix variables of (55) is (3r2 + 6)n2 + 3n with n = 3 and r = 8.

CR IP T

Since the TTSD technique reduces the complexity of fuzzy modeling of the original nonlinear error system, the control design problem under consideration can be solved with a less computational burden. With the resulting sampled-data fuzzy controller (18) from Theorem 1, the angular velocity in the fast subsystem can track the controllike reference angular velocity command of the form (4) generated by the slow subsystem quickly enough. However, Theorem 1 only presents the exponential stability and H∞ tracking performance of the closed-loop of the fast subsystem (19); nothing about the closed-loop of the original nonlinear error system (2) is discussed. Next, we will show that the proposed controller in Theorem 1 can exponentially stabilize the original nonlinear error system with an H∞ tracking performance if the control-like parameters K1σ and K2σ are chosen appropriately.

 ˜ ˜ ˜T ˜  PA + A P  ∗   ∗  ∗

AN US

Theorem 3. Suppose that for given scalars γ > 0, a and b satisfying (24), r1 , r2 , and r3 , matrices P¯ > 0, X¯ 1 = X¯ 1T , ¯ K¯ j , and M ¯ i j , i, j ∈ S such that the LMIs (41) hold. If for the chosen parameters K1σ X¯ 2 , R¯ > 0, S¯ l > 0, l = 1, 2, W, and K2σ , there exists a common matrix P˜ > 0 satisfying the LMIs as follows: 0 ¯ li j Π ∗ ∗

P˜ B˜ 0 ¯T 0 µI¯ Tl W −I 0 ∗ −I

    < 0 

(56)

M

¯ li j , l = 1, 2, i, j ∈ S are denoted in (41), I¯ 1 = where A˜ is defined in (6), B˜ = [03 I3 ]T , µ = maxt∈[t0 , t f ] kG(σ(t))k, Π [I1 03 03 03 03 03 ], and I¯ 2 = [I¯ 1 03 ], then, the sampled-data fuzzy controller (18) can stabilize exponentially the ˆ = 0 and achieve the H∞ tracking performance in (23) with P given by original nonlinear error system (2) with d(t) ˆ (43) in presence of d(t).

CE

4. Simulation Study

PT

ED

Proof: See Appendix B.  It is seen from Theorem 3 that the validity of the proposed H∞ sampled-data fuzzy control design relies on the choices of K1σ and K2σ . If they are appropriately chosen (i.e., there exists a common matrix P˜ > 0 so that the LMIs (56) hold), then the designed H∞ sampled-data fuzzy controller can exponentially stabilize the original nonlinear error system with an H∞ tracking performance, which implies that it can satisfactorily achieve the tracking of the attitude command generated by the guidance system of MEV.

AC

In this section, simulation results are given to illustrate the effectiveness of the developed method. The parameter values and the initial attitude conditions of the MEV are given as follows: J xx = 2983kg · m2 , Jyy = 4909kg · m2 , Jzz = 5683kg · m2 h iT 0.1581 −0.1482 σ(0) = 0.3637 h iT 0.200 0.2000 . ω(0) = 0.2000

In our numerical simulation, the reference attitude command comes from the results in [40], and the corresponding MRPs are shown in Fig. 2. For the slow subsystem, the matrices K1σ and K2σ in (4) are selected as K1σ = diag{3, 3, 3}, K2σ = diag{0.02, 0.02, 0.02}. 13

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V d1 Vd2 Vd3

0.4

Vd

0.2 0 -0.2 0

50

100 150 time (sec.)

200

250

CR IP T

Figure 2. The reference attitude command σd (t).

Assume that the angular velocities ω1 (t) ∈ [−4, 4], ω2 (t) ∈ [−4, 4], and ω3 (t) ∈ [−4, 4]. Then, the membership functions in (12) are rewritten as follows: ωg (t) + 4 8 4 − ωg (t) µg,2 (ωg (t)) = , g = 1, 2, 3. 8

AN US

µg,1 (ωg (t)) =

CE

PT

ED

M

Next, the optimization problem (55) is solved to obtain the sampled-data fuzzy controller (18) for the system (16). Suppose ug,max = 6000, g = 1, 2, 3, dˆmax = 2000, and ν = 5000. Let C N1 = 80I, DN2 = 0, r2 = 0, and h = 0.2. By using the program fminsearch, the optimal scaling parameters are obtained as r1 = 1526.1485, r3 = 463.4976, ε = 0.00122113, a = 0.2247, and b = 0.7034 under the initial values r1 = 1600, r3 = 400, ε = 0.001, a = 0.2, and b = 1. Then, the solution of the optimization problem (55) with the optimal scaling parameters is derived. Using this solution, we can get the gain matrices      −8.1285 1.3958 −2.5041   −6.8630 1.6617 −2.5745      K1 = 103 ×  3.1554 −8.2413 −0.8702  , K2 = 103 ×  2.8177 −7.5094 0.9080      3.7389 3.2358 −7.0211 0.3786 −1.6633 −7.3306      −7.7912 2.1230 2.6951   −8.1807 −0.3808 −1.9285      K3 = 103 ×  3.3243 −6.9964 −0.7498  , K4 = 103 ×  −3.2615 −8.2456 −1.1636      −4.2114 −2.2555 −6.8983 3.6675 −2.6361 −7.5781      −6.4104 −0.4554 2.6366   −6.3610 −0.6290 −2.3774      K5 = 103 ×  −3.3369 −8.8305 −0.9186  , K6 = 103 ×  −3.2901 −8.9223 0.9539      −0.3654 −0.9903 −7.1603 0.1874 1.0864 −7.3348      −6.9056 1.6043 2.7168   −8.3878 −0.6527 1.9097      K7 = 103 ×  3.1552 −8.3310 1.1567  , K8 = 103 ×  −3.8176 −7.6831 1.5241  (58)     −0.6062 1.8060 −7.0812 −4.8165 3.4141 −7.1295

AC

√ and γopt = ρ¯ = 118.9334. Now, we apply Theorem 3 to ensure the validity of the TTSD method. With the chosen K1σ , K2σ in (57), and the solution in the optimization problem (55), we can verify that there exists a common matrix P˜ > 0 satisfying the LMIs in (56). Thus, the chosen parameters K1σ and K2σ are suitable for the TTSD. Choose the following external disturbance     0.25 sin(2t) exp(−0.6t)   d(t) =  0.25 sin(2t + π/4) exp(−0.6t)  .   0.25 cos(2t + π/2) exp(−0.6t)]

For the convenience of demonstration, we only select the first 10 seconds trajectory in the simulation to validate the effectiveness of the designed strategy. To show the advantage of the resulting fuzzy controller, four other types of controllers are given for comparison. Controller I: a continuous-time fuzzy controller 14

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With the same K1σ and K2σ in (57), the following continuous-time fuzzy controller for the system (16) is given in [19]: u(t) = τ(t) − B−1 (

8 X i=1

ϑi (ω(t))Ai ωd (t) − ω ˙ d (t))

K1 = diag{15845, 15820, 15602} K2 = diag{23768, 23730, 23403} K3 = diag{31691, 31641, 31204} K4 = diag{39613, 39551, 39005} K5 = diag{47536, 47461, 46806} K6 = diag{55459, 55371, 54607} K7 = diag{63381, 63281, 62408}

AN US

K8 = diag{71304, 71191, 70209}.

CR IP T

where τ(t) = −K j eω (t), K j ∈ R3×3 . We choose the same γ = 118.9334 and apply Theorem 1 in [19]. The following control gains can be obtained:

(59)

Controller II: a sampled-data proportional-integral (PI) controller With the same K1σ and K2σ in (57), we select the following sampled-data PI controller for the system (16): Z tk u(t) = K1ω eω (tk ) + K2ω eω (s)ds, t ∈ [tk , tk+1 ), k ∈ N (60) 0

αPˆ −1GT (σe )σe ˆ −1 ˆ − R P(ωe + k1 σe ) 1 + σTe σe

(61)

ED

u(t) = −

M

where K1ω = −diag{4000, 4000, 4000} and K2ω = −diag{30, 30, 30}. Controller III: a nonlinear controller A nonlinear controller is given as follows [48]:

PT

where α = 10, k1 = 0.15, Pˆ = diag{200, 200, 200}, and Rˆ = diag{0.02, 0.02, 0.02}. Controller IV: a finite-time controller We choose the following finite-time control law [10]: u(t) = −k1 (

1 + eT e )J(v p + k2p e)2/p−1 − s(ω)Jω + JRbd ω ˙ d + Js(v)Rbd ωd 4

(62)

AC

CE

where p = 75 , k1 = 35, and k2 = 2. With Controllers I and IV, one can get a good tracking performance while the control constraints are not satisfied. However, for given the sampling interval h = 0.2 and the gains K j , j ∈ S in Controller I, we can find that the attitude tracking error system is unstable in Fig. 3. By using Controllers II–IV with the sampling interval h = 0.2 and the proposed controller in this paper, simulation results are shown in Figs. 4–6, which imply that the proposed sampleddata fuzzy controller can get a better tracking performance. Furthermore, we can obtain |ug (t)| < 4000, g = 1, 2, 3, R 10 −2 ˆ ˆ dˆT (s)d(s)ds 6 0.0816 < γopt dmax = 0.1414, and t 0

Z

10

t0

(eTω (s)N1 eω (s) + uT (s)N2 u(s))ds = 891.1812

2 6 eTω (t0 )Peω (t0 ) + γopt

Z

t0

10

ˆ dˆT (s)d(s)ds ≈ 4153.6741

2 6 x˜ (t0 )P˜ x˜(t0 ) + eTω (t0 )Peω (t0 ) + γopt T

15

Z

t0

10

ˆ dˆT (s)d(s)ds ≈ 4153.6757



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     / Information Sciences 00 (2018) 1–23 16  which mean that the control constraints (22) and the tracking performances (20) and (23) are ensured. Moreover, the  also indicate that the existing continuous-time controllers cannot efficiently deal with the attitude simulation results  tracking sampled-data control design for MEV.  to further test the effectiveness   Subsequently, of the proposed method, a 60-run Monte Carlo simulation is per  formed with random dispersions in the initial angular velocity. The final attitude tracking error keσ (10)k2 is shown in −3   Fig. 7. As expected, the range of keσ (10)k 2 is [0, 3.5 × 10 ]. 

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Remark 6. With the gains (58), the proposed sampled-data fuzzy controller for attitude tracking of MEV generates torque commands, which can be implemented through the reaction control system (RCS) (see e.g., [26, 27]). From Fig. 6, we can obtain |ug (t)| < 4000, g = 1, 2, 3 and need more RCS jets to achieve the attitude tracking of MEV. Note that we can adjust some parameters in the optimization problem (55) to obtain small control gains (for examples, select some small bounds ug,max , g = 1, 2, 3 or a large disturbance attenuation level γ). Such a parameter adjustment will influence the determination of the controller gains and the tracking performance. Generally, there is a tradeoff among the high gain, solvability and convergence. How to solve this issue will be addressed in future research activities. 



eV

t 

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eV  t eV  t eV  t

 



  





  WLPHVHF









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Figure 3. The attitude tracking error eσ (t) under Controller I with sampled-data.

AC

CE

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 V  V d  

V  V d 

V  V d 

Figure 4. σd (t) (black lines), σ(t) under Controller II (green lines), under Controller III with sampled-data (blue lines), under the proposed method in this paper (red lines).

5. Conclusion In this study, the attitude tracking problem of MEVs with control constraints has been studied via an H∞ sampleddata fuzzy control approach. Based on the TTSD technique, the original nonlinear error system is initially divided into a fast subsystem describing the attitude dynamics and a slow subsystem describing the attitude kinematics. Subsequently, the DIC method is employed to obtain the angular velocity command for the slow subsystem. Then, based on 16

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V 3 (V d 3 )

    

V 2 (V d 2 )

V 1 (V d 1 )

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2000

0

u (t )

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Figure 5. σd (t) (black lines), σ(t) under Controller IV with sampled-data (green lines), under the proposed method in this paper (red lines).

u1 (t ) u2 (t ) u3 (t )

-2000

-4000 2

4 6 time (sec.)

8

10

M

0

Figure 6. The control input u(t) using the proposed method in this paper.

CE

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the angular velocity command and the T-S fuzzy model of the fast subsystem, an H∞ sampled-data fuzzy controller design with control constraints is proposed in terms of LMIs for the tracking error fuzzy system on the basis of a time-dependent functional. The resulting controller can ensure the exponential stability and H∞ tracking performance of the closed-loop of the original nonlinear error system, if a suitable timescale separation between the fast and slow subsystems is given. Finally, simulation results have illustrated the effectiveness of the developed method. Our future works will extend the developed approach to the attitude tracking problem of MEVs with observer-based sampled-data control or fault-tolerant sampled-data control. Acknowledgments

AC

This work was supported in part by the National Natural Science Foundation for Distinguished Young Scholars of China under Grant 61625302, in part by the National Natural Science Foundations of China under Grants 61473011, 61721091, and 61803177, in part by National Basic Research Program of China (973 Program) (2012CB720003), and in part by Shandong Provincial Natural Science Foundation under Grant ZR2018BF015. The authors gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation of this paper. Appendix A The following lemma will be employed in the proof of Theorem 1. Lemma A1: Consider the disturbance-free fuzzy system of (19), we have the relation as follows: keω (t)k 6 θ0 keω (tk )k, t ∈ [tk , tk+1 ). 17

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10

5

0

0

0.5

1

1.5

2

2.5

3

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Number of Cases

15

3.5

4

−3

x 10

Figure 7. The final attitude tracking error keσ (10)k2 .

V˙ 0 (t) =

8 X 8 X i=1 j=1

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where θ0 > 0 is a scalar. ˆ = 0, we obtain for t ∈ [tk , tk+1 ) that Proof: Set V0 (t) = keω (t)k2 . Along the solution of the system (19) with d(t) ϑi (ω(t))ϑ j (ω(tk ))

×[2eTω (t)Ai eω (t) + 2eTω (t)BK j eω (tk )]

6 2kAkkeω (t)k2 + 2kBkkKkkeω (t)kkeω (tk )k 6 θ1 V0 (t) + θ2 V0 (tk )

M

where θ1 = 2kAk + kBkkKk and θ2 = kBkkKk in which kAk = max kAi k, kKk = max kK j k, i, j ∈ S. It follows that i

j

h  i V0 (t) 6 eθ1 (t−tk ) + (θ2 /θ1 ) eθ1 (t−tk ) − 1 V0 (tk )

AC

CE

PT

This completes the proof. Proof of Theorem 1: Define

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for t ∈ [tk , tk+1 ). Based on tk+1 − tk 6 h, we obtain h  i V0 (t) 6 eθ1 h + (θ2 /θ1 ) eθ1 h − 1 V0 (tk ), t ∈ [tk , tk+1 ).



¯ , W −1 , P , W T PW ¯ P , diag{W T , W T , W T }L, W ¯ Xl , W T X¯ l W, S l , W T S¯ l W, l = 1, 2, R , W T RW

¯ i j W, i, j ∈ S K j , K¯ j W, Mi j , diag{W T , W T , W T } M

Λ1 , diag{W T , W T , W T , W T , I, I, I}

Λ2 , diag{W T , W T , W T , W T , I, W T , I, I}.

Pre- and post-multiplying both sides of (41) with Λl and its transpose, respectively, and using Schur complement, we get the LMIs as follows: Πli j < 0, l = 1, 2, i, j ∈ S

18

(A1)

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where Π1i j

Π2i j

  ij j  Φ111 (h) + Υ j Φi112 (h) PB    =  ∗ −vS 1 0    ∗ ∗ −γ2 I   ij j  Φ211 (h) + Υ j + hMi j R−1 MiTj Φi212 (h) PB    =  ∗ −vS 2 0   2  ∗ ∗ −γ I

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j j (h) = Ωi20j + hΩi21j (h) (h) = Ωi10j + hΩi11j (h), Φi112 Φi111 j j Φi211 (h) = Φi10j + hΩi12j (h), Φi212 (h) = Ωi20j + hΩi22j (h)

in which v is defined in (25). Notice that Ξi j (t, h) = 1h (ψ(t)Π1i j + (h − ψ(t))Π2i j ). Using (A1), it is immediate that 8 X 8 X

ϑi (ω(t))ϑ j (ω(tk ))Ξi j (t, h) < 0.

i=1 j=1

(A2)

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Obviously, the inequality (A2) means that there exists a small enough scalar ¯ satisfying the inequality (39) for any  ∈ (0, ¯ ). Thus, we have the inequality (40). From (40), it follows that Z t ˆ (eTω (s)N1 eω (s) + uT (s)N2 u(s) − γ2 dˆT (s)d(s))ds tk

6 V f (tk ) − V f (t), t ∈ [tk , tk+1 ).

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For any t0 < t f ∈ S(h), there exists an integer k1 ∈ N+ such that t f = tk1 . Since V f (t) is continuous in [t0 , t f ) and V f (tk−1 ) > 0, it can be seen explicitly that Z tf ˆ (eTω (s)N1 eω (s) + uT (s)N2 u(s) − γ2 dˆT (s)d(s))ds t0

ED

=

kX 1 −1 Z tk+1 k=0

tk

ˆ {eTω (s)N1 eω (s) + uT (s)N2 u(s) − γ2 dˆT (s)d(s)}ds

6 V f (t0 ) − V f (t1− ) + V f (t1 ) − V f (t2− ) + · · · + V f (tk1 −1 ) − V f (tk−1 )

PT

= V f (t0 ) − V f (tk−1 )

6 V f (t0 ) = eTω (t0 )Peω (t0 )

AC

CE

which implies that

Z

tf

t0

6

(eTω (t)N1 eω (t) + uT (t)N2 u(t))dt

eTω (t0 )Peω (t0 )

+γ

2

Z

tf

ˆ dˆT (t)d(t)dt.

t0

In addition, it is clear from (39) that 8 X 8 X

¯ <0 ϑi (ω(t))ϑ j (ω(tk ))Σi j (t, h)

i=1 j=1

which implies that there exists a positive scalar κ satisfying the inequality as follows: 8 X 8 X i=1 j=1

¯ 6 −κI ϑi (ω(t))ϑ j (ω(tk ))Σi j (t, h) 19

(A3)

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where 1 (ψ(t)Γ1i j + (h¯ − ψ(t))Γ2i j ) h¯ " ij # ¯ Φi j (h) ¯ Φ111 (h) 112 = ∗ −vS 1 " ij j ¯ # ¯ ¯ (h) Φ211 (h) + hMi j R−1 MiTj Φi212 . = ∗ −vS 2

¯ = Σi j (t, h) Γ1i j Γ2i j

D+ V f (t) 6

8 8 X X

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ˆ = 0, and considering (A3), we have Thus, from (35)–(37) with d(t)

¯ η(t) ϑi (ω(t))ϑ j (ω(tk ))η˜ T (t)Σi j (t, h) ˜

i=1 j=1

6 −κη˜ T (t)η(t) ˜ 6 −κξT (t)ξ(t) where η(t) ˜ = [ξT (t) φ(t)˜eTω (t)]T . Next, let us define

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¯ 0] ξ(t0 + θ) , ϕ(θ), θ ∈ [−h,

(A4)

(A5)

where ϕ(θ) denotes an artificial initial condition constructed by ( ξ(t0 ), θ = 0 ϕ(θ) = ¯ 0). 0, θ ∈ [−h,

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Considering that ψ(t) and φ(t) are bounded on [t0 , +∞), it follows from (33) that there exist scalars α0 > 0 and α1 > 0 so that Z t V f (t) 6 α0 kξ(t)k2 + α1

Choose κ0 > 0 such that

t−h¯

kξ(s)k2 ds.

(A6)

By (A4) and (A6), we have

ED

¯ κ0 h¯ ) − κ 6 0. κ0 (α0 + α1 he

(A7)

PT

D+ (eκ0 t V f (t)) 6 eκ0 t {(κ0 α0 − κ)kξ(t)k2 + κ0 α1

Z

t

t−h¯

kξ(s)k2 ds}.

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Integrating the above inequality gives κ0 t

AC

e V f (t) 6 e

κ 0 t0

V f (t0 ) +

Z

t

t0

2

κ0 s

e {(κ0 α0 − κ)kξ(s)k + κ0 α1

¯ κ0 h¯ ) − κ) 6 eκ0 t0 V f (t0 ) + (κ0 (α0 + α1 he ¯

¯ κ0 h +κ0 α1 he

Z

t0

t0 −h¯

Z

t0

t

Z

s s−h¯

kξ(τ)k2 dτ}ds

eκ0 s kξ(s)k2 ds

eκ0 s kξ(s)k2 ds.

(A8)

Then, using (A5) and (A7), it follows from (A8) that eκ0 t V f (t) 6 eκ0 t0 V f (t0 ), i.e., V f (t) 6 e−κ0 (t−t0 ) V f (t0 ). By (34), we can get from (A9) that keω (tk )k 6

p κ0 λ2 /λ1 e− 2 (tk −t0 ) keω (t0 )k 20

(A9) (A10)

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where λ1 = λmin (P) and λ2 = λmax (P). Moreover, there exists an integer k0 ∈ N so that t ∈ [tk0 , tk0 +1 ) for any given t > t0 . Thus, we obtain from Lemma A1 and (A10) that p κ0 keω (t)k 6 θ0 keω (tk0 )k = θ0 λ2 /λ1 e− 2 (tk0 −t0 ) keω (t0 )k p κ0 κ0 6 θ0 λ2 /λ1 e 2 h keω (t0 )ke− 2 (t−t0 ) 

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which means that the disturbance-free fuzzy system of (19) is exponentially stable over S(h). Therefore, the tracking error fuzzy system (19) is exponentially stable with γ-disturbance attenuation. Appendix B

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Proof of Theorem 3: From (2), (6), and (19), the original closed-loop nonlinear error system is derived as follows:  ˜  x˙˜(t) = A˜ x˜(t) + BG(σ(t))e  ω (t)      8 8 XX (B1)   ˆ  e˙ ω (t) = ϑi (ω(t))ϑ j (ω(tk ))[Ai eω (t) + BK j eω (t − ρ(t))] + Bd(t)     i=1 j=1 For system (B1), a functional candidate is selected as

V(t) = V s (t) + V f (t)

(B2)

where V s (t) and V f (t) are defined in (7) and (33), respectively. Notice that µ = maxt∈[t0 , t f ] kG(σ(t))k. Along the solution of this system, we have

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V˙ s (t) = 2 x˜T (t)P˜ x˙˜(t) ˜ A˜ x˜(t) + BG(σ(t))e ˜ = 2 x˜T (t)P( ω (t))

6 2 x˜T (t)P˜ A˜ x˜(t) + x˜T (t)P˜ B˜ B˜ T P˜ x˜(t)

ED

+eTω (t)GT (σ(t))G(σ(t))eω (t) ˜ x˜(t) + µ2 eTω (t)eω (t). 6 x˜T (t)(P˜ A˜ + A˜ T P˜ + P˜ B˜ B˜ T P)

(B3)

For t ∈ [tk , tk+1 ), we obtain from (35)–(37) and (B3) that

AC

where

CE

PT

ˆ D+ V(t) + eTω (t)N1 eω (t) + uT (t)N2 u(t) − γ2 dˆT (t)d(t) ˜ x˜(t) + µ2 eTω (t)eω (t) 6 x˜T (t)(P˜ A˜ + A˜ T P˜ + P˜ B˜ B˜ T P) 8 X 8 X ¯ + ϑi (ω(t))ϑ j (ω(tk ))ηT (t)Ξi j (t, h)η(t) i=1 j=1

6 ζ T (t)zζ(t)

(B4)

h iT I˜ = [I1 03 03 03 ], ζ(t) = x˜T (t) ηT (t) " # P˜ A˜ + A˜ T P˜ + P˜ B˜ B˜ T P˜ 0 z= ¯ + µ2 I˜ T I˜ ∗ Ξi j (t, h)

¯ is defined in (38). in which Ξi j (t, h) h iT ˆ = 0 can be represented Moreover, letting e˜ (t) = x˜T (t) eTω (t) , the original nonlinear error system (B1) with d(t) by ¯ e(tk ) ¯ e(t) + B˜ e˙˜ (t) = A˜ 21

(B5)

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where   A˜  A¯ =   0

˜ BG(σ(t)) 8 P ϑi (ω(t))Ai

i=1

   0   ¯  , B =  0 

   8 P . ϑ j (ω(tk ))BK j  0

j=1

Similarly in Lemma A1, we obtain that the error system (B5) has the property as follows: ¯ e(tk )k, t ∈ [tk , tk+1 ) k˜e(t)k 6 θk˜

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where θ¯ is a positive scalar. Assume that the LMIs (56) are satisfied. Then, applying an analogous analysis in the proof of Theorem 1, we get that the system (B1) is exponentially stable and H∞ tracking performance in (23) is achieved with P given by (43), which means that the sampled-data fuzzy controller (18) can exponentially stabilize the original nonlinear error system (2) with an H∞ tracking performance.  References

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