Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2013, Article ID 429101, 9 pages http://dx.doi.org/10.1155/2013/429101

Research Article

A Second-Order Sliding Mode Controller Design for Spacecraft Tracking Control

Yu-Xin Zhao, Tian Wu, and Gang Li

College of Automation, Harbin Engineering University, Harbin 150001, China Correspondence should be addressed to Yu-Xin Zhao; zhaoyuxin@hrbeu.edu.cn Received 21 July 2013; Accepted 19 August 2013 Academic Editor: Rongni Yang

Copyright © 2013 Yu-Xin Zhao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For spacecraft attitude tracking system, there exists the chattering phenomenon. In this paper, the spacecraft motion is decomposed into three-channel subsystems, and a second-order sliding mode control is proposed. This method has been proved to have good convergence and robustness. Combined with the proposed sliding surface, the three-channel controllers are designed. The control performance is confirmed by the simulation results, the approaching process is improved effectively, and a smooth transition is achieved without overshoot and buffeting.

1. Introduction

Sliding mode variable structure control (SMC) is very suitable for spacecraft control system design because of its unique advantages, such as the simple algorithm, no need of the precise mathematical model for the controlled object, and the invariability of the control system when the parameter perturbation exists and the external disturbance conforms exactly to the matching condition [1].

But classical variable structure control system has a fatal flaw, namely "chattering" phenomenon, which is the main obstacle for SMC to promote in the engineering field [2]. In order to eliminate or suppress the chattering, scholars have conducted a wide-ranging and in-depth study [3-5]. In the past, the solution was to use the continuous function instead of the discontinuous parts, which had played a certain role in weakening buffeting, but it also greatly reduced the control performance [6]. A variable structure terminal guidance law with adaptive quick-reaching law and a filter approach based on sliding mode observer was developed [7], and the simulation results demonstrated higher accuracy, better robustness, and easy realization of the designed guidance law. For the guidance of attacking target on the ground, a new robust nonlinear sliding mode guidance law with terminal angle constraint was proposed [8]. Wu et al. introduced some slack matrices and a delay-dependent sufficient condition to guarantee the sliding mode dynamics is generalized, quadratically

stable, and robustly passive [9,10]. And Wu et al. designed an integral sliding surface function and an observer to estimate the system states, which was proved to be attained in a finite time [11, 12]. An integral-type sliding surface function is designed for establishing a sliding mode dynamics, which can be formulated by a switched stochastic system with an external disturbance/uncertainty [13]. But the control methods used are based on a general variable structure control theory, which is very unfavorable for the normal operation of spacecraft control system due to the existence of the chattering phenomenon; it could easily lead to system instability. And in the application of general variable structure control theory, the controlled object must be a relative degree 1. When the controlled object relative degree is greater than 1, this theory cannot be used.

During the flight, the aerodynamic parameters change intensely, and the coupling between three channels is very serious, which is caused by the flight speed, dramatically changed air density, and other disturbances. It brings about great difficulties because the entire spacecraft control system presents a strong multivariable coupling, time-varying, non-linearity and uncertainty. The traditional method of spacecraft control system design is based on the assumption of small perturbation and the coefficient of frozen [14,15]. However, the large angle of aircraft maneuvering is inevitable. Therefore, it is difficult to reconcile three-channel controller design based on the classical control theory. BTT control

technology is a future trend in this field, but it would make the coupling aggravate [16-18]. In the design of spacecraft control system, even if the spacecraft is regarded as a particle, the relative motion between spacecraft and target is strongly nonlinear [19, 20].

Due to the reasons above, it will encounter enormous difficulties to use classical control theory to design the spacecraft control system in future [21, 22]. Therefore, it is necessary to search a high-level sliding variable structure control method for the spacecraft control system, which cannot only keep strong robustness but also can achieve continuous control [4, 23-26]. Shtessel et al. proposed the second-order sliding mode variable structure control algorithm, which was proved to be good robustness and to eliminate effectively the chattering [27-29]. A second-order sliding mode controller is designed [30] for the aerodynamic missile, which was simple and effective. The results of the simulation indicated that this second-order sliding mode controller is robust for the systems with uncertainties and can effectively reduce the chatting.

However, there are certain limitations to the high-order sliding mode control algorithm, such as the demanding controlled object and the complexity of algorithm. The paper conducts a new research on the second-order sliding mode variable structure control system, and simulation results show that the method we proposed can effectively eliminate the chattering phenomenon and have good robustness in the spacecraft control systems. The research results have injected new idea for the sliding mode variable structure control algorithm and also have provided a new solution for the spacecraft control system design. Therefore, the study of this subject has not only theoretical significance but also has a very important application reference value.

In this paper, we are concerned with the chattering phenomenon in the spacecraft attitude tracking system. The spacecraft motion is decomposed into three-channel subsystems, and a second-order sliding mode control is proposed, which has good convergence and robustness. Combined with the proposed sliding surface, the three-channel controllers are designed. Furthermore, the control performance is confirmed by the simulation results, the approaching process is improved effectively, and a smooth transition is achieved without overshoot and buffeting. The rest of this paper consists of five sections about reaching condition and controller design. The spacecraft model is described in Section 2, and a three-channel decomposition model is proposed. Section 3 proposes a second-order sliding mode. Section 4 designs the three-channel controller. Section 5 validates control performance. Finally, we conclude this paper in Section 6.

2. Spacecraft Model

The spacecraft attitude motion equations introduced in [31] is considered here. The model are described as follows:

Jx~7T = MX -{Jz -Jy)UzUy>

dt dwy

^ =My-(JX-Jt

dw. t \

=Mz -(]y -]*)WxWr

9 = toy sin y + toz cos y,

f=-^-(toy cos y-to. sin y), cosy v 7 7

y = tox - tan 9 (toy cos y - toz sin y) ,

where the pitch angle 9, the yaw angle f, and the roll angle y are referred to the posture angle. tox, toy, and to. are the angular velocity of the spacecraft around the shaft Ox, Oy, and Oz. Jx, Jy, and Jz are the moments of inertia, Mx, My, and Mz are the external torques on the body frame.

From (1), we can see that the spacecraft attitude motion equation is nonlinear. And because of the existence of the large attitude angle, the coupling between three channels cannot be ignored.

2.1. Decomposition of Mathematical Model. In order to make the process of decomposition clear, (1) can be broken down into three subsystems.

(i) Channel Subsystem of Pitch Angle. The state function for the channel subsystem of pitch angle can be obtained by Figure 1.

From Figure 1, we can obtain the following state function:

x1z 0 1 x1z

X2z. 0 0 X2z.

0 [fz\

where uz = Mz, which is the controller of pitch channel subsystem.

And = toy sin y + toyj cos y - (cos yUz)(Jy - Jx^toy -tozy sin y is the coupling between the channels, which can be regarded as the external interference.

(ii) Channel Subsystem ofYawAngle. The state function for the channel subsystem of yaw angle can be obtained by Figure 2. From Figure 2, we can obtain the following state function:

X1y 0 1 X1y

.X2y. 0 0 X2y.

cos y cos 9JV

where uy = My is the controller of yaw channel subsystem, and

cos 9J.

, , cosy

Uz - Jx) + -

cos v sin 99 - sir

. sin y cos y cos 9y + sin y9 sin 9

can be regarded as the external interference.

z) z x>

(iii) Channel Subsystem of Roll Angle. The state function for the channel subsystem of roll angle can be obtained by Figure 3.

1 o *2zJ LO

X1 z x2z

Figure 1: The state function for the channel subsystem of pitch angle.

From Figure 3, we can obtain the following state function:

X1x 0 Ï X1x

.X2x. 0 0 X2x.

where ux = Mx is the controller of roll channel subsystem, and

f = ^y -Jx

wzwy - sec

0-0 (wy cos y - wz sin y)

- tan 9 (wy cos y - wyy sin y - wz sin y - wzy cos y)

can be regarded as the external interference.

The process of deduction regards the coupling between three channels as the external interference. When the spacecraft carries a maneuver of large angle attitude, the coupling will be exacerbated, which will affect the work of three subsystems seriously. Therefore, in the design of controller, we should make the control system have better robustness.

3. Reaching Condition Design and Proof

The system (2) can be expressed as:

x(t) = A(x)+B(x)u + f(x), (7)

where xeRn, ueRm, A e Rn, B e Rnxm, and B(x) = (b1, b2, ...,bm); each component of the vector function A(x) and bt (i = 1,..., m) are sufficientlysmooth scalar functions. f(x) is the external interference. The phase trajectoryofthe system (2) will be significantly affected due to the existence of f(x).

3.1. Model System Description and Simplification. Assume that f(x) = 0, and the system description can be simplified as

x (t) = A(x) + B (x) u.

Designing the sliding mode surface a = x1t due to the theory [32] of second-order sliding mode, we can get the following function:

a = x2, ö = a(x) + b (x) u.

3.2. Reaching Conditions. Define the following reaching condition:

s = -k1s - k2s - ejsj" sat (s),

r Start

From equation (1), we get the expression of y/

y = ((Oy cos y-(0z silly)

Calculate the time derivative of 1p

Jy-^j- = My- (Jx - Jz)a>zwx

cos y Sin i

V = -Q^y + -

■ cos y - cos tf • sin y ■ y Sin V Sin {

-—-— L®v--irö)z--

cos 8 y cos a

• sin y + cos 8 • cos y ■ y cos2 8

Take state vector Xly = f. x2y = f

.. cos y cos v , „ cos y sin 88 - sin yy cos 9 . sin y cos y cos 9y + sin y9 sin 8

V = + TTzrU* - + —---»V - «k—£----L^rJL-«k

cos B]y y cos 9Jy

cos2 8

-co., - --

y cos 8

cos2 8

Figure 2: The state function for the channel subsystem of yaw angle.

where kt > 0, k2 > 0, e > 0, a > 1, and sat(-) indicates the saturation function, which is used to eliminate chattering

sat (s) =

1 s > A ks |s| < A -1 s < -A

Theorem 1. By satisfying the reaching condition (10), thephase trajectory of the system (8) can reach the set points Q = {x | i(x) = s(x) = 0} at a limited time and stay at this state.

Proof. Define that h(s) = k2s + e|s|a sat(s), and we can get sh(s) > 0. Here, we select the Lyapunov function below

V = 0.5s2 +

\ h(y)dy J0

and V > 0. The time derivative of formula (12) can be obtained as

V = s (-k1s - k2s - e|s|a sat (s)) + h (s) s. (13) By substituting h(s) = k2s + e|s|a sat(s) into (13), V can hold

V = -kxs . Next, we will make some analysis about V.

(i) When s = 0, V < 0; that is to say that V ^ 0. And in order to make lim|s|^+TO JQ h(y)dy = 0, there must be limt^+TOs = 0 for an arbitrary initial position of the system.

(ii) When s = 0, s = 0. At this time, h(s) = k2s + e|s|a sat(s) = 0. Because of sh(s) > 0, s = 0 is the only zero for this function. Therefore, s = 0.

According to (i) (ii) above, s will be zero in the end, and the phase trajectory of system (8) will reach the set points Q = {x | s(x) = s(x) = 0} asymptotically and stay at this state. □

Explanation. In the reaching condition s = -k1s - k2s -e|s|a sat(s) above, the reaching speed will become faster when k2s+e|s|a sat(s) getsbigger,butthephase trajectorywillresuilt in larger chattering in neighborhood of s = 0. However, if we make k2s + e|s|a sat(s) smaller, the reaching time will become longer with the decrease of speed. Besides, the reaching speed will slow down if the coefficient k1 is too large, and the chattering in neighborhood of s = 0 will be weakened in this condition. On the contrary, if the coefficient k1 is too small, the reaching speed will enlarge and the phase trajectory will result in larger chattering in neighborhood of s = 0.

f Start

From equation (1), we get the expression of y

Y = «>x - tan S(ioy cos y - wz sin y)

Calculate the time derivative of y

y = aix - sec S ■ 9(wy cos y - u>z sin y) - tan d(coy cos y - wyy sin y - wz sin y - a>zy cos y)

Take state vector *ix = r> *2x = V

y = — Mx + —--WziOy - sec2 S • d(iüy cos y - coz sin y) - tan d(tOy cos y - wyy sin y - d>z sin y - wzy cos y)

Xlx X2x

0 o\[X2x

C End J

Figure 3: The state function for the channel subsystem of roll angle.

3.3. Robustness Prove. Combining (9) with (10), we can get a(x) + b(x)u = -k1s - k2s - eH" sat(s), and the expression of controller can hold as

u = b-1 (x) [-k1s - k2s - e|s|a sat (s) - a (%)]. (15)

Here, the equivalent external interference is considered, namely, f(x) = 0. Then, we can get

a = %2,

a = a(x) + b (x)u + f (x).

Thus, by substituting (15) into (16), the sliding mode surface can be formulated as

s = -k1s - k2s - e|s|a sat (s) + f (x). (17)

Assumption 2. f(x) is bounded, there exists |f| < M, i = 1,.. .,m.

Theorem 3. By satisfying the Assumption 2 and e > M (i = 1,2,..., m), the phase trajectory of the system (7) can reach the set points Q = jx | i(x) = s(x) = 0} under the control law (15) and stay at this state.

Proof. Define that h(s) = k2s + e|s|a sat(s) - f, and we can get sh (s) = k2s2 + e|s|"+1 - fs > k2s2 + e|s|"+1 - Ms

= k2s2 + (e|s|a -M) s.

(i) When |s| > tfMfe, sh(s) > 0.

(ii) When |s| < rfM/e, |s|a+1 ^ 0, a ^ 1, k2 > e.

sh(s)>k2s2 -Ms = k2 (^-)-M>0. (19)

Here, we select the Lyapunov function below

V = 0.5s2 + \ h(y) dy. (20)

The process of proof is the same as Theorem 1. Then, we can get the conclusions of V = -k1s2, and limt^+TOs = 0. □

From Theorems 1 and 3, formula (17) can be regarded as the system controller regardless of the equivalent external interference if only it meets Assumption 2. So, the robustness of the controller can be reflected. Theoretically, we can make k1 and e large enough to achieve the better robustness and dynamic performance. However, in practical system, we will encounter limited energy of the controller; thus, it is impossible to make k1 and e infinite.

4. Controller Design

Combined with the proposed second order sliding mode control law (10), we can get the following controllers.

4.1. Controller Design of Pitch Channel. Assume that pitch angle 9 tracks a given angle 9d and 9d is second-order derivative. Meanwhile, the equivalent external interference fz is bounded.

Let = [x1z x2z] , and in order to use the sliding mode variable structure control of second-order conveniently, we select sz = e = x1z - 9d. And then, we can get the following functions:

S, = X-

'z = x2z - ®d>

q = cos Yu q

-Vd = —j—uz - Vd-'z

(21) (22)

And if sz is regarded as the output of system (2), its relative order is 2, which satisfies the condition of Theorem 1. Then, we select the reaching condition as follows:

*z = -klzéz - k2zsz - yr sat (Sz) ,

where k1z > 0, k2z > 0, and ez > 0.

Combining formula (22) with formula (23), we can get

-k1z$z - k2zsz - ezlsr sat (Sz) = ——Uz - 9d. (24)

Therefore,

($d - k1zéz - k2zsz - sat (^z)) • (25)

From the deduction above, we know that sz will converge to zero in a finite time. That is to say that the pitch angle 9 will be on track for the given angle 9d in a limited time.

4.2. Controller Design of Yaw Channel. The process of deduction is the same as pitch channel. For the system (3), assume that the yaw angle f tracks a given angle fd and fd is second-order derivative. Meanwhile, the equivalent external interference fy is bounded.

Let xy = [x1y x2y] , and we select sy = e = x1y - fd. And then, we can get the following functions:

Sy = X2y - fd'

Sy = X2y - fd

cos 9Jy y

Uy - fd•

(26) (27)

Regarding sy as the output of system (3), we select the reaching condition as follows:

Sy = -kiySy - k2ysy - £y\s\a sat (sy) , (28)

where kly > 0, k2y > 0, and ey > 0.

Combining formula (27) with formula (28), we can get

cos 9JV

■ (9d - kXyèy - k2ySy - eyl^r sat (sy)) • (29)

Thus, sy will converge to zero in a finite time. That is to say that the yaw angle f will be on track for the given angle fd in a limited time.

4.3. Controller Design of Roll Channel. For the system (5), assume that the roll angle y tracks a given angle yd and yd is second-order derivative. Meanwhile, the external outside interference fx is bounded.

Let = [X1X X2X]T, and we select = e = x^ - yd. Then, we can get the following functions:

Sx = X2x Yd' 1

Sx = X2x - Yd = YUx - Yd•

Regarding sX as the output of system (5), we select the reaching condition as follows:

h = -k1x$x - k2xSx - £x|s|" sat (sX) ' (31)

where k1x > 0, k2x > 0, and ex > 0. And we can get

Ux = Jx (Yd - k1x$x - k2xSx - exM" sat (sx)). (32)

Thus, sx will converge to zero in a finite time. That is to say that the yaw angle y will be on track for the given angle yd in a limited time.

In the spacecraft control system, we want roll angle to calm to zero, and in order to achieve target tracking, generally, we let the pitch angle and yaw angle track a given angle.

5. Simulation Results

In this section, experimental simulations will be carried out to evaluate the effectiveness of the proposed sliding mode controller of second order:

x1z 0 1 0 0 0 0 x1z

x2z 0 0 0 0 0 0 X2z

X1y 0 0 0 1 0 0 X1y

X2y 0 0 0 0 0 0]] X2y

X1x [[0 0 0 0 0 1 X1x

X2x . [0 0 0 0 0 0] . X2x .

cos X1x cos XizJy 0 0

0 0 0 0

Detailed parameters of this control system are presented as follows. The initial angles are [9 f y] = [n/6 n/3 n/4]; the angular accelerations are [0 y y] = [0.2 0.4 0.5].And the desired tracking angles are [9d fd Yd] = [n/4 n/6 0]. The related parameters of system are Jx = 224 kg • m2, Jy = 284 kg • m2, and Jz = 324 kg • m2. The equivalent interference

fz = fy = fx = 3 sin t. Then, formula (33) can be reduced as follows:

x1z 0 1 0 0 0 0 x1z

x2z 0 0 0 0 0 0 X2z

X1y 0 0 0 1 0 0 X1y

X2y 0 0 0 0 0 0 X2y

X1x 0 0 0 0 0 1 X1x

X2x . 0 0 0 0 0 0 . X2x .

3 sin t

3 sin t

3 sin t

284 cos x.

0 0 0 0

S- — —

The three sliding surface are selected as sz = xlz - n/4, sy = xly - n/6, and sx = xlx. Firstly, we assume that fz = fy = fx = 0, and we can get the following functions:

Sz = X2z' (35)

cos x.

Sy — X2z >

y 2z 284 cos ^ y

lx 224"x And we select the same reaching conditions:

h = -8s z - 5s z - SW sat (sz),

Sy = -8Sy - 5sy - 5|syp sat (sy) , (41)

^x = -8ix-5sx - |Sx|" sat (sx)-

By taking formulas (36), (38), (40), and (41), the control law of each channel can hold as

284 cos x.

Uy —

(-8sz - 5Sz - 5|sz\a sat (sz)) , (-8s y - 5s y - 5|syp sat (sy)),

0 2 4 6

- Pitch angle

---Yaw angle

----Roll angle

10 12 14 16 18 20 Time (s)

Figure 4: The angle tracking of the three subsystems.

0.8 0.6 0.4 0.2 0 -0.2 -0.4 -0.6

— 224(-8èx-5Sx -|sx|a sat (Sx)) .

0 2 4 6 8 10 12 14 16 18 20 Time (s)

Figure 5: The switching function curve of the three subsystems.

Under the control law (42), the phase trajectory of three-channel will reach the sliding mode in a finite time. Simulation diagrams are as shown in Figures 4, 5, and 6.

From Figure 4, we can see that the proposed control law achieves a better angle tracking, and no overshoot occurs. From Figure 5, it can be seen that the proposed sliding surface achieves a smooth transition in the whole process of approaching, and there is no chattering. By Figure 6, we can see that chattering of the controllers reduces significantly during the approaching process under a certain external interference.

Time (s)

Figure 6: The control output of the three subsystems.

6. Conclusion

A three-channel model of spacecraft has been analysed in this paper. By simplifying the model, a second-order sliding surface was proposed, and its conditions of convergence and robustness were analyzed. Combined with the proposed sliding surface, three-channel controllers were designed, respectively. The control performance was confirmed by the simulation results, the approaching process was improved effectively, and a smooth transition was achieved without overshoot and buffeting. The coupling between the three channels was fully taken into account in the design of the controller, and it was equivalent to external interference, which provided a high practical value.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (nos. 60904087 and 51109045), and the Fundamental Research Funds for the Central Universities of China (nos. HEUCF110419 and HEUCFX41302).

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