Disturbance observer based control for flexible spacecraft with time-varying input delay
© Zhang et al.; licensee Springer 2013
Received: 18 April 2013
Accepted: 7 May 2013
Published: 22 May 2013
In this paper, a composite disturbance-observer-based control (DOBC) and control scheme is applied to solve the spacecraft attitude control with time-varying input delay. Compared with some existing results, distinct features of the proposed method are that the delay-dependent disturbance observer and controller are used to estimate and compensate the main disturbance caused by flexible appendages and to attenuate exogenous bounded disturbances with attenuation level, respectively. The proposed design is obtained by combining the augmented Lyapunov functional with linear matrix inequality technique. The effectiveness of the proposed design method is illustrated via a numerical example.
Flexible spacecraft plays an important role in communication, remote sensing and a variety of space related research works [1–4]. However, the design of these appendages often involves the need for large, complex and light-weight space structures to achieve increased functionality at a reduced launch cost . During the control of the rigid body attitude, the unwanted excitation of flexible modes, together with other external disturbances, measurement and actuator error, may degrade the performance of attitude control systems . On the other hand, spacecrafts usually operate in the presence of various disturbances, which include radiation torque, gravitational torque, aerodynamic torque and non-environmental torques, etc. . The problem of disturbance rejection is particularly pronounced in the case of low-earth-orbiting satellites that operate in altitude ranges where their dynamics are substantially affected by most of the disturbances mentioned above [2, 8]. In modern control theory, anti-disturbance control methodologies can be divided into two main types. One is the disturbance attenuation method such as control [9–11]. The other is the disturbance rejection method which may compensate the disturbance via the compensator [12–14]. However, these approaches only deal with one type of disturbance. In practice, together with the rapid development of sensor and data processing technologies, the disturbances or noise from different sources (e.g., sensor and actuator noise, friction, vibration) can be characterized by different mathematical models. Also, disturbance can represent the unmodeled dynamics and system uncertainties. For the case of multiple disturbances, a composite hierarchical anti-disturbance control was proposed [7, 15] to guarantee the simultaneous disturbance attenuation and rejection performance.
In practical application, input delay always exists in a flexible spacecraft due to the physical structure and energy consumption of the actuators. Although it is not the most important factor to affect the attitude control, it still leads to substantial performance deterioration and even to instability of the system [16, 17]. Hence, anti-disturbance control algorithms for such systems that explicitly take input time delay into account are of practical interest. Up to now, the issue of anti-disturbance control problems for flexible spacecraft subject to both disturbance and input time delay has not been fully investigated and remains to be open and challenging.
Motivated by the preceding discussion, in this paper, a composite attitude controller design approach is designed for a flexible spacecraft based on DOBC and state-feedback control. By constructing an augmented Lyapunov functional with slack variables, new delay-dependent DOBC and controller are obtained in terms of linear inequality matrices. The resultant DOBC can reject the effect of vibrations from flexible appendages, and state-feedback control can attenuate the influence of the norm bounded disturbances. Moreover, compared with the existing results [6, 7, 16], (I) Input time delay is considered in the designed controller, which is more practical than the methods in [6, 7]; (II) An augmented Lyponov function is used to design controller, which may lead to a more relaxed design than the methods in . Finally, a numerical example is shown to demonstrate the good performance of our method.
Notation: Throughout this paper, denotes the n-dimensional Euclidean space; the space of square-integrable vector functions over is denoted by ; the superscripts `⊤’ and ‘−1’ stand for matrix transposition and matrix inverse, respectively; means that P is real symmetric and positive definite (semidefinite). In symmetric block matrices or complex matrix expressions, stands for a block-diagonal matrix, and ∗ represents a term that is induced by symmetry. For a vector , its norm is given by . Matrices, if their dimensions are not explicitly stated, are assumed to be compatible for related algebraic operations.
Problem formulation and preliminaries
Meanwhile, it is noted that belongs to and , where . For the system described by (7)-(8), the objectives of this paper are as follows:
Designing the observer gain L and controller gain K makes the closed-loop system (7)-(8) asymptotically stable;
The performance of system (7)-(8) satisfies for any nonzero under the zero initial condition.
To obtain our main results, we need the following lemma.
Lemma 1 
In this section, we consider the augmented system (7)-(8). We give the design method based on LMI to compute the controller gain and the observer gain simultaneously.
where and . According to (21) and (22), follows from (9a)-(9b), which implies that holds for any nonzero . □
In Theorem 1, a new Lyapunov-Krasovskii functional is constructed by employing slack variables and . It is noted that and are useless for reducing the conservatism of stability conditions in . However, they can provide a more relaxed design of the controller later on since need only be an invertible matrix rather than a positive definite matrix.
On the basis of Theorem 1, we will present a design method of the disturbance observer based controller in the following.
From (9a)-(9b), it is clear that (23a)-(23b) holds. As a result, the closed-loop system (7)-(8) is asymptotically stable and satisfies . The proof is thus completed. □
Compared with the design method in , the matrices and are invertible matrices instead of positive definite matrices, which makes the design more flexible. Moreover, the augmented Lyapunov functional method also can be extended to the systems without time delay.
For given , , , and , solve LMI (23a)-(23b) with , , , , , , , ;
Compute K and L through , ;
Construct the controller and observer as (4) and (6).
In this paper, composite disturbance-observer-based control (DOBC) and control scheme has been investigated. The LMI-based conditions are formulated for the existence of the admissible disturbance observer and controller, which ensures that the closed-loop system is asymptotically stable with a disturbance attenuation level. A numerical simulation shows the performance of the attitude control system. Further improvement in a composite disturbance observer with output feedback control for flexible spacecrafts will be considered in our future work.
This work was supported in part by the Major State Basic Research Development Program of China (973 Program) under Grant No. 2012CB720003, in part by the National Science Foundation of China under Grant No. 91016004, 60904025, in part by Qing Lan project of Jiang Su province and in part by the scholarship from China Scholarship Council.
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