## Advanced Stochastic Control Systems with Engineering Applications

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# Relative Orbit Stabilization Control for the Agile Satellite under Stochastic Disturbance

**Academic Editor:**Ming Liu

#### Abstract

This paper investigates the relative orbit control problem for a space communication satellite network. An observer-based state feedback control scheme is developed under the circumstance of faults and disturbance occurring in the sensors and actuators. The validity of sliding mode observer for the satellites’ network is deduced and the analysis and proof of the relative orbit stabilization control are completed.

#### 1. Introduction

The agile satellites have brought a great number of conveniences for modern spatial application; they have evolved from single satellite to constellation and formation [1]; furthermore, a spatial dynamic network is constructed. In order to expand the range of imaging services, orbit maneuver ability becomes an intrinsic ability of the satellite, and the satellites do not run on their preselected orbit. Therefore, the relative position and velocity need to be measured; all the members in the spatial agile imaging network need to be controlled real-timely [2].

The research of relative motion control is focusing on the two aspects, namely, relative orbit control and relative attitude control. Some methods are proposed for relative orbit control, such as optimal control for the orbit rendezvous [3] and predictive control for the rendezvous maneuver [4]. The relative control methods include the centralized and decentralized approach [5]. These methods pay more attention to the control law design in an ideal world where the status measurement sensors and control actuators work well. The modern control method is mentioned and analyzed [6–11] while, in the practical orbit control, the sensors and actuators will work in the fault or disturbance status.

This paper will complete the studies and analysis of sliding mode observer and state feedback control based on designed observer. The relative motion dynamic model will be depicted in Section 2; the sliding mode observer will be discussed in Section 3; the state feedback controller will be designed and its stabilization analysis is completed in Section 4.

#### 2. Dynamic Model of Network Members

The relative motion dynamics of satellite communication network is usually established in the local-vertical-local horizontal [LVLH] coordinate system [12]. The relative motion equation can be rewritten as in the following form:

Formula (1) can be expanded into the following form:

The , , and are the relative coordinates to the target spacecraft, is the orbit angular of the target spacecraft, is the orbit radius of the spacecrafts, , , is the control acceleration, is the gravitational constant, and . Actually (2) is equivalent to

Hence , , and can be derived, and the expressions, respectively, are

Consider each expression of variables in the system (4):

We define augmented variable as follows:

The system (4) can be rewritten as in the following form:

Aiming at system (7), we consider more complex and practical situation: there exist sensors and actuators fault, constant input disturbance (namely, the last item in formula (7)), and the output disturbance in the system. Then we define system matrix as follows:

The system (7) can be rewritten as Here, , , , , and represent system matrix and and , respectively, are actuators fault and sensors fault. is the sensors’ disturbance. is the constant disturbance. Here we pay attention to a more general situation: the disturbance meanwhile exists in the state equation and output of system (9).

Defining , we do the following assumptions to the system (9).(A1)fault and the perturbation vector: , , , and satisfy the following assumption: where , , , , , , and are known constant.(A2)() is able to be observed, and there exists constant , which makes (A3)matrix: are column full rank matrix.

For the convenience of discussion, we define augmented vector and matrix as follows:

Establish an augmented generalized system based on system (9) as follows:

Matrix and matrix have the following properties:

Therefore, it can be inferred that, according to matrix knowledge, there must be an appropriate number of dimensions matrix which makes an invertible matrix. We may define a new matrix where and , , . Meanwhile, we define a new matrix . We can calculate directly

We can conclude that there must be according to (16). Suppose that has the following form:

In formula (17), , , , , and are matrixes needed to be solved. Expand as follows: Consider the elements in the last line of the expression . Consider Formula (20) can be obtained by directly calculating

#### 3. Observer Design

To get the asymptotic estimates of the state of system (9) and, at the same time, solve the corresponding control problem, we introduce sliding-mode observer as follows:

Here, where , , , , , , , , , . , , and , respectively, are derivative gain, proportional gain, and sliding gain of the observer and is defined previously. The and are not real estimation of and in the observer (21). Assume that the real estimations of and , respectively, are and ; thus, According to the assumption (A3), and are column full rank, so and exist. It can be concluded from (23) that

Lemma 1. *In the case of (A2), for the observer (21), there is a gain matrix , which makes Hurwitz.*

*Proof. *First, considering matrix , for the finite dimensions matrix, there must exist a constant , making , (), which means , ().

So, for arbitrary real number , the following matrix rank relationship existed:
In formula (26), notice that, for any , always existed. Therefore,
On the other hand, we notice that

So we can derive . According to (26), we can derive the following formula:

Then we discuss the values of and , based on formula (29).

First, we consider the situation of and, at this time, formula (29) can be equivalent to

We can draw the conclusion based on the assumption (A2)

On the other hand, consider the condition of and, at this moment, formula (28) turns into
Notice that is column full rank and the above formula turns into
Integrating the above two cases derived, we have proved that, for any , . Hence, is a couple observed, and can elicit that can be observed. Therefore, there exists matrix , making Hurwitz (i.e., the eigenvalues of are all negative). Thus there must exist matrix which makes

Let the proportion gain of observer (21) be equal to ; then, we can calculate

Therefore,, . The proof completes.

#### 4. Observer Error System

Then we derive the error system of the observer (21). Firstly, in the first formula of the system (13), we add at the left and right side and we can get

On the other hand, for the observer (21), we can get Notice that , so . Therefore, from formula (36), we can obtain We define error variable as follows: Meanwhile considering (36) and (38), we obtain This is equal to In formula (41), Here, Hence, if the value of is large enough, in the system (41), will become infinitesimal. Until now, we remove the influence of disturbance for the system stability.

On the other hand, for the of the system (41), we design as the following form ( is design parameter): of which, is Lyapunov matrix required and satisfies Here, is matrix parameters waiting for being solved. Based on the above analysis, error system (41) changes as

The following section discusses the stability of error system (46) and then discusses stabilization problem of closed-loop system.

#### 5. The Stability Analysis of the Error System

Theorem 2. *For error system (46), let the sliding mode observer gain , if there exists matrix making the following matrix inequality established:
**Then the system (46) states trajectory asymptotically stable convergence to the origin.*

*Proof. *For the system (46), defining Lyapunov function , , along the system (46) state trajectory, we can calculate
Consider parts of above formula,

The following formation can be derived based on assumption (A1):
Therefore
So we can derive

If. the inequality always holds. So the error system (46) is asymptotically stable. The proof completes.

#### 6. The Stabilization of Closed-Loop System

Now we consider the stabilization problem of the closed-loop system based on the observer. Considering the system (9), we design a state feedback controller based on observer as follows:

Substitute formula (53) into the system (9) and we can obtain Here, , .

Formula (54) can be rewritten as Here, .

For the close-loop system (55) and the error system (46), they can construct the following system:

We present the following theorem.

Theorem 3. *If there is symmetric positive definite matrix and the matrix , which makes the following constraint matrix established:
**
then the system (56) is asymptotically stable.*

*Proof. *For the system (56), we define Lyapunov function

Here, is a positive definite symmetric matrix waiting for being solved. Along the system (56) trajectory, we can directly calculate

Let and then

Here, represents the minimum eigenvalue of the matrix . We define a new Lyapunov function

Here, is a parameter waiting for design, is defined in Theorem 2.

According to the proof in Theorem 2, we can obtain

Here,

In addition, the new parameters are defined as follows:

Selecting parameters and let it satisfy

Calculate formula (61) further and we can obtain

Notice that , , so we have

So we have proved that the system (56) is asymptotically stable.

#### 7. Conclusions and Future Works

The sliding mode observer and the state feedback controller is proposed and the controller’s stabilization under stochastic disturbance is proved. The research provides a theoretical analysis of the controller design method based on sliding mode observer. In the future, we will give the simulation verification combined with the specific space mission.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

This work is partially funded by the Fundamental Research Funds for the Central Universities (no. HEUCF021318), the Natural Science Foundation of Heilongjiang Province (no. A201312), the National Natural Science Funds (no. 11372080), the Harbin Science and Technology Innovation Talent Youth Fund (no. RC2013QN001007), and the Key Laboratory of Database and Parallel Computing, Heilongjiang Province.

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#### Copyright

Copyright © 2014 Xiande Wu 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.