Sampled-Data Control of Spacecraft Rendezvous with Discontinuous Lyapunov Approach

Abstract

This paper investigates the sampled-data stabilization problem of spacecraft relative positional holding with improved Lyapunov function approach. The classical Clohessy-Wiltshire equation is adopted to describe the relative dynamic model. The relative position holding problem is converted into an output tracking control problem using sampling signals. A time-dependent discontinuous Lyapunov functionals approach is developed, which will lead to essentially less conservative results for the stability analysis and controller design of the corresponding closed-loop system. Sufficient conditions for the exponential stability analysis and the existence of the proposed controller are provided, respectively. Finally, a simulation result is established to illustrate the effectiveness of the proposed control scheme.

1. Introduction

In the research area of advanced astronautic missions, spacecraft autonomous rendezvous and docking has been one of the most significant problems over the past few decades. Generally speaking, before completing the final docking action between the chaser spacecraft and the target spacecraft, the chaser spacecraft is forbidden to maintain at a fixed relative position around the target and prepares to implement the final docking action and/or other special missions [1–3]. This procedure is the so-called “relative position holding.” In recent years, a few results have been developed in the existing literatures to investigate the problems of autonomous relative motion for multiple spacecrafts [4, 5]. Among these existing approaches, one of the most popular mathematical tool is the so-called Clohessy-Wiltshire (C-W) equation. This model is introduced by Clohessy and Wiltshire in 1960 [6] and has been broadly applied to address the problem of relative motion between two neighboring spacecrafts [7, 8].

On the other hand, in modern industrial process, one or more computers are always employed as digital controllers to control continuous-time plant [9–12]. Under this setting, digital computers are required to sample and quantize a continuous-time analog signal to evaluate a discrete-time sampled-data signal [13–15], based on which a discrete-time control input signal is designed [16–19]. This discrete-time signal is further converted into a continuous-time signal based on a zero-order hold equipment in the actuator side [20–24]. Subsequently, the corresponding sampled-data systems contain both continuous-time and discrete-time signals in the continuous-time domain [25–28]. Due to its significance, over the past decade the problems of stabilization and filtering of sampled-data systems have been investigated extensively; see for instance, [29, 30] and the reference therein.

It is worth pointing out that there exist three main methodologies for the stabilization and filtering problems of sampled-data systems [31, 32], which are lifting technique method, impulsive model method, and input delay method, respectively. Among these existing approaches, input delay approach is the latest established approach, wherein the sampled-data system is modeled as a continuous-time system with time-varying delay existing in the control input. A great number of results based on this approach have been proposed in the existing literatures [33, 34]. Among these existing results, the time-independent Lyapunov function approach presented by Fridman et al. [34] has been proved to be effective to deal with the input delay. Based on the time-independent Lyapunov approach, an improved Lyapunov approach, namely, time-dependent Lyapunov approach has been also developed by Fridman [33] and Naghshtabrizi et al. [35]. Compared with the time-independent Lyapunov approach, the structure characteristic of the time-dependent Lyapunov approach can guarantee that its value does not increase in each sampling time despite the jumping behavior of the sampled signals and thus can reduce the design conservativeness. This method is later extended to deal with filtering problem of Itô stochastic systems with signal sampling and quantization [36].

It should be pointed out that, in practical sampled-data systems of spacecraft rendezvous, there also exists jumping behavior of the sampled signal at each sampling time instant, and the values of the sampled signal increase abruptly at these time instants. Therefore, it is desirable to reduce the conservativeness of the controller design of sampled-data spacecraft rendezvous systems by developing new control schemes. Motivated by the aforementioned time-dependent Lyapunov approach, in this paper, we consider the sampled-data control problem for relative position holding of spacecraft rendezvous. In this study, signal sampling, control input limitation, and 𝐻_∞ performance are considered together in a unified framework. The discontinuous Lyapunov functions approach is adopted for the controller design; also we introduce free-weight matrices in the Lyapunov functions, and it will be shown that the obtained results are less conservative then those in the existing work.

The remainder of this paper is organized as follows. In Section 2, the problem of sampled-data control of relative position holding of spacecraft rendezvous is formulated. In Section 3, the sufficient condition for the exponential stability of the sampled-data closed-loop system is provided, based on which the sufficient for the existence of the sampled-data controller is given. Finally, a numerical example is presented in Section 4 to demonstrate the feasibility of the proposed method.

2. Problem Formulation

In the orbital coordinate frame of chaser and target spacecrafts, it is well defined that the origin attaches to the mass center of the target spacecraft, 𝑥-axis is along the vector from the Earth center to the origin, 𝑦-axis is along the target orbit circumference, 𝑧-axis completes the right-handed frame, 𝑟₁ denotes the radius of the target circular orbit, 𝜇 denotes the gravitational parameter of the earth, 𝑤 denotes the angle velocity of the target, and 𝑤=(𝜇/𝑟³₁)^1/2.

Based on the Newton motion theory, the relative dynamic model of chaser and target spacecrafts can be modeled by the following C-W equations: 𝑥̈_𝑐(𝑡)=2𝑤𝑦̇_𝑐(𝑡)+𝑤²𝑟₁+𝑥_𝑐(𝑡−𝜇𝑟₁)+𝑥_𝑐(𝑡)𝑟₁+𝑥_𝑐(𝑡)²+𝑦²_𝑐(𝑡)+𝑧²_𝑐(𝑡)^3/2+𝑎_𝑥,𝑦̈_𝑐(𝑡)=−2𝑤𝑥̇_𝑐(𝑡)+𝑤²𝑦_𝑐(𝑡)−𝜇𝑦_𝑐(𝑡)𝑟₁+𝑥_𝑐(𝑡)²+𝑦²_𝑐(𝑡)+𝑧²_𝑐(𝑡)^3/2+𝑎_𝑦,𝑧̈_𝑐(𝑡)=−𝜇𝑦_𝑐(𝑡)𝑟₁+𝑥_𝑐(𝑡)²+𝑦²_𝑐(𝑡)+𝑧²_𝑐(𝑡)^3/2+𝑎_𝑧, (1) where 𝑥_𝑐(𝑡), 𝑦_𝑐(𝑡), and 𝑧_𝑐(𝑡) are the components of the relative position in corresponding axes and 𝑎_𝑖 (𝑖=𝑥,𝑦,𝑧) is the 𝑖th component of the control input vector. By the taylor expansion and linearization approach, the linearized equation around the null solution for (1) can be given by 𝑥̈_𝑐(𝑡)−2𝑤𝑦̇_𝑐(𝑡)−3𝑤²𝑥_𝑐(𝑡)=1𝑚𝑇_𝑥+𝑑_𝑥,𝑦̈_𝑐(𝑡)+2𝑤𝑥̇_𝑐(𝑡)=1𝑚𝑇_𝑦+𝑑_𝑦,𝑧̈_𝑐(𝑡)+𝑤²𝑧_𝑐(𝑡)=1𝑚𝑇_𝑧+𝑑_𝑧. (2) In system (2), 𝑚 is the mass of the chaser, 𝑇_𝑖 (𝑖=𝑥,𝑦,𝑧) is the relative motion dynamic, and 𝑑_𝑖, 𝑖=𝑥,𝑦,𝑧 is the 𝑖th component of the external disturbances. We define the following vector and matrices: 𝑥(𝑡)=𝑥_𝑐(𝑡)𝑦_𝑐(𝑡)𝑧_𝑐(𝑡)𝑥̇_𝑐(𝑡)𝑦̇_𝑐(𝑡)𝑧̇_𝑐(𝑡)(𝑡)=𝑇,𝑢^𝑥(𝑡)𝑇_𝑦(𝑡)𝑇_𝑧(𝑡),𝑑(𝑡)=𝑑_𝑥(𝑡)𝑑_𝑦(𝑡)𝑑_𝑧(𝑡)(𝑡)=𝑥,𝑦^𝑐(𝑡)𝑦_𝑐(𝑡)𝑧_𝑐(𝑡),𝐴=0001000000100000013𝑤²0002𝑤0000−2𝑤0000−𝑤²,1000𝐵=𝑚.000000000100010001 (3)(4) Then the C-W equation (2) can be described equivalently as follows: 𝑥̇(𝑡)(𝑡)(𝑡)(𝑡),𝑦(𝑡)(𝑡).=𝐴𝑥+𝐵𝑢+𝐵𝑑=𝐶𝑥 (5)

As a result, the specific relative motion of the chaser and target can be realized via designing effective control strategy 𝑢(𝑡) for system (5).

In this paper, we consider the case that the signals are sampled before transmitted to the controller side. Define 𝒯≜{𝑡₁,𝑡₂,𝑡₃,…} as a strictly increasing sequence of sampling times in (𝑡₀,∞) for initial time 𝑡₀=0. Suppose that the sampling internals between any two sequent sampling instants are bounded by 𝑡_𝑘+1−𝑡_𝑘≤ℎ,𝑘=1,2,…, (6) where ℎ>0 is a known constant. Then, the control vector is assumed to be of the following form at the sampling instant 𝑡_𝑘 with zero-order hold (ZOH): 𝑢(𝑡)=𝐾𝑥𝑡_𝑘. (7) During the orbital transfer process, the thrust constraint can be described as 𝑢_𝑖𝑡_𝑘⩽𝑢_𝑖,max𝑖=𝑥,𝑦,𝑧, (8) where 𝑢_𝑖(𝑡_𝑘) is the control input thrust along the 𝑖th axis at the sampling instant 𝑘 and 𝑢_𝑖,max is the maximum allowed thrust along the 𝑖th axis.

On the other hand, the desired holding position can be treated as a reference output 𝑦_𝑟 of (5). Therefore, our object is thus to propose a sampled-data control strategy (7), such that the closed-loop system for plant (5) is exponentially stable and the output 𝑦(𝑡) tracks the reference signal 𝑦_𝑟 without steady-state error. Subsequently, the problem of relative control holding is converted into an output tracking control problem. To achieve lim_𝑡→+∞𝑦(𝑡)=𝑦_𝑟, (9) we define the output error 𝑒(𝑡)=^𝑡₀𝑦_𝑒(𝑡)𝑑𝑡. (10) Also, we define the following augmented system 𝜉̇(𝑡)=𝐴𝜉(𝑡)+𝐵𝑢𝑡_𝑘+𝐵₁𝑑(𝑡) (11) with 𝜉(𝑡)=𝑥(𝑡)𝑒(𝑡),𝑑(𝑡)=𝑑(𝑡)𝑦_𝑟,𝐴=𝐴0_6×3𝐶0_3×3,𝐵0𝐵=_3×3,𝐵₁=𝐵0_6×30_3×3−𝐼_3×3,𝐶₁=𝐶0_3×3,𝐶₂=0_3×6𝐼_3×3. (12) The control scheme for augmented system (11) is thus designed as 𝑢𝑡_𝑘=𝐾𝜉𝑡_𝑘=𝐾𝐾_𝑒𝑥𝑡_𝑘𝑒𝑡_𝑘, (13) where 𝐾 is defined as in (2) and 𝐾_𝑒∈ℝ^𝑚×𝑛 is to be designed. Substituting (13) into (11) yields the following augmented closed-loop system: 𝜉̇(𝑡)=𝐴𝜉(𝑡)+𝐵𝐾𝜉𝑡_𝑘+𝐵₁𝑑(𝑡),𝑦(𝑡)=𝐶₁𝜉(𝑡),𝑧(𝑡)=𝐶₂𝜉(𝑡). (14) If system (14) is exponentially stable, then it implies that the relative position error integral action 𝑒(𝑡)→0 and further implies that lim_𝑡→∞=𝑦_𝑟.

In the following discussion, we adopt the delay-input approach to deal with the sampled-data control problem. By defining 𝜏(𝑡)=𝑡_𝑘+1−𝑡_𝑘, (15) the sampling intervals can be written as 𝑡_𝑘=𝑡−𝑡_𝑘+1−𝑡_𝑘=𝑡−𝜏(𝑡). (16) Therefore, the sampled augmented state can be written as 𝜉𝑡_𝑘=𝜉(𝑡−𝜏(𝑡)), (17) and the control input vector can be transformed into 𝑢𝑡_𝑘=𝐾𝜉𝑡_𝑘=𝐾𝜉(𝑡−𝜏(𝑡)). (18) Hence, the augmented closed-loop system (14) can be written by 𝜉̇(𝑡)=𝐴𝜉(𝑡)+𝐵𝐾𝜉(𝑡−𝜏(𝑡))+𝐵₁𝑑(𝑡),𝑦(𝑡)=𝐶₁𝜉(𝑡),𝑧(𝑡)=𝐶₂𝜉(𝑡). (19) Now, the aim of this paper can be formulated as follows.

Problem 1. Design a sampled-data control law 𝑢(𝑡_𝑘), such that the following conditions are met. (1)The closed-loop system (19) is exponentially stable with a given decay rate 𝛼>0. In other words, the chaser holds at a relative position. (2)The control input satisfies the given upper bound (8). (3)For closed-loop system (19), the following 𝐻_∞ performance index is achieved: 𝑒_𝑧(𝑡)=‖𝑧(𝑡)‖₂−𝛾𝑑(𝑡)₂⩽0. (20)

3. Main Results

In this section, we will first perform the stability analysis and control scheme design for system (19) with 𝑑(𝑡)=0. Then, we will consider the internal stability of (19) with the prescribed 𝐻_∞ performance index (20).

3.1. Stabilization Analysis

Theorem 2. Consider the closed-loop system (19) with 𝑑(𝑡)=0, given a decay rate 𝛼>0, if there exist positive definite matrices 𝑃₁, 𝑃₂, and 𝑃₃∈ℝ^9×9, and matrices 𝑇₁, 𝑇₂, 𝑆₁, 𝑆₂, and 𝑆₃∈ℝ^9×9, such that the following conditions hold: ΦΦ=¹¹Φ₁₂Φ₁₃⋆Φ₂₂Φ₂₃⋆⋆Φ₃₃Φ<0,Φ=¹¹Φ₁₂Φ₁₃Φ₁₄⋆Φ₂₂Φ₂₃0⋆⋆Φ₃₃Φ₃₄⋆⋆⋆Φ₄₄𝜌<0,𝐾^𝑇𝑅^𝑇_𝑖𝑅_𝑖𝐾−𝑢²_𝑖,max𝑃₁<0, (21)(22) where 𝜌 is defined as in (47) and Φ₁₁=2𝛼𝑃₁−𝑃₃+2𝛼ℎ𝑃₃−𝑇^𝑇₁𝐴−𝐴^𝑇𝑇₁−𝑆₁−𝑆^𝑇₁,Φ₁₂=𝑃₁+𝑇^𝑇₁−ℎ𝑃₃,Φ₁₃=𝑃₃−2𝛼ℎ𝑃₃−𝑇^𝑇₁𝐵𝐾+𝑆^𝑇₁,Φ₂₂=ℎ𝑃₂+𝑇₂+𝑇^𝑇₂,Φ₂₃=−ℎ𝑃₃−𝑇^𝑇₂𝐵𝐾+𝑆₂+𝑆^𝑇₂,Φ₃₃=−𝑃₃+2𝛼ℎ𝑃₃+𝑆₃+𝑆^𝑇₃,Φ₁₁=2𝛼𝑃₁−𝑃₃−𝑇^𝑇₁𝐴−𝐴^𝑇𝑇₁−𝑆₁−𝑆^𝑇₁,Φ₁₂=𝑃₁+𝑇^𝑇₁,Φ₁₃=𝑃₃−𝑇^𝑇₁𝐵𝐾+𝑆^𝑇₁,Φ₁₄=ℎ𝑆^𝑇₁,Φ₂₂=𝑇^𝑇₂+𝑇^𝑇₂,Φ₂₃=−𝑇^𝑇₂𝐵𝐾+𝑆₂+𝑆^𝑇₂,Φ₃₃=−𝑃₃+𝑆₃+𝑆^𝑇₃,Φ₃₄=ℎ𝑆^𝑇₃,Φ₄₄=−ℎ𝑒^−2𝛼ℎ𝑃₂. (23) Then system (19) with 𝑑(𝑡)=0 is exponentially stable with the decay rate 𝛼>0.

Proof. For system (19), consider the following Lyapunov functional 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡), (24) where 𝑉₁(𝑡)=𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡),𝑉₂(𝑡)=(ℎ−𝜏(𝑡))^𝑡_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^{−2𝛼(𝑡−𝑠)}𝑉𝑑𝑠,₃(𝑡)=(ℎ−𝜏(𝑡))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉𝑡_𝑘−𝜉(𝑡). (25) In function (3), 𝑃_𝑖>0  (𝑖=1,2,3) are Lyapunov matrices to be designed, and 𝛼>0 is the decay rate as previously defined. It can be calculated that 𝑉̇₁(𝑡)+2𝛼𝑉₁(𝑡)=2𝜉̇^𝑇(𝑡)𝑃₁𝜉(𝑡)+2𝛼𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡),𝑉̇₂(𝑡)=−𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠+((𝑡𝑒𝑑𝑠ℎ−𝜏))^−2𝛼𝑡()×−2𝛼_𝑡𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠=((𝑡𝑒𝑑𝑠ℎ−𝜏))^−2𝛼𝑡×𝜉̇^𝑇(𝑡)𝑃₂𝜉̇(𝑡)𝑒^2𝛼𝑡−𝜉̇^𝑇𝑡_𝑘𝑃₂𝜉̇𝑡_𝑘𝑒^{2𝛼𝑡𝑘} (26) Notice that 𝜉̇^𝑇(𝑡_𝑘)𝑃₂𝜉̇(𝑡_𝑘)𝑒^{2𝛼𝑡𝑘}=0; thus, 𝑉̇₂(𝑡)=−𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠((𝑡𝑒𝑑𝑠−2𝛼ℎ−𝜏))^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠+((𝑡𝜉̇𝑑𝑠ℎ−𝜏))^𝑇(𝑡)𝑃₂𝜉̇(𝑡). (27) Therefore, it is derived that 𝑉̇₂(𝑡)+2𝛼𝑉₂(𝑡)=−𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠((𝑡𝑒𝑑𝑠−2𝛼ℎ−𝜏))^−2𝛼𝑡×^𝑡_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠+((𝑡𝜉̇𝑑𝑠ℎ−𝜏))^𝑇(𝑡)𝑃₂𝜉̇(𝑡)+2𝛼(ℎ−𝜏(𝑡))𝑒^−2𝛼𝑡×^𝑡_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠𝑑𝑠=−𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠+((𝑡𝜉̇𝑑𝑠ℎ−𝜏))^𝑇(𝑡)𝑃₂𝜉̇(𝑡). (28) Furthermore, for 𝑉₃(𝑡) it is calculated directly that 𝑉̇₃(𝑡)=−𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉̇−𝜉ℎ−𝜏))^𝑇(𝑡)𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉𝑡−𝜉ℎ−𝜏))_𝑘−𝜉(𝑡)^𝑇𝑃₃𝜉̇(𝑡),𝑉̇₃(𝑡)+2𝛼𝑉₃(𝑡)=−𝜉𝑡_𝑘−𝜉(𝑡)^𝑇×𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉̇−𝜉ℎ−𝜏))^𝑇(𝑡)×𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡×−𝜉ℎ−𝜏))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉̇(𝑡)+2𝛼(ℎ−𝜏(𝑡))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇×𝑃₃𝜉𝑡_𝑘−𝜉(𝑡). (29) As a result, we have 𝑉̇(𝑡)+2𝛼𝑉(𝑡)=𝑉̇₁(𝑡)+2𝛼𝑉₁(𝑡)+𝑉̇₂(𝑡)+2𝛼𝑉₂(𝑡)+𝑉̇₃(𝑡)+2𝛼𝑉₃(𝑡)⩽2𝜉̇^𝑇(𝑡)𝑃₁𝜉(𝑡)+2𝛼𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡)−𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑒^2𝛼𝑠+((𝑡𝜉̇𝑑𝑠ℎ−𝜏))^𝑇(𝑡)𝑃₂𝜉̇(𝑡)−𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇×𝑃₃𝜉𝑡_𝑘(𝑡−((𝑡𝜉̇−𝜉)ℎ−𝜏))^𝑇(𝑡)×𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉𝑡−𝜉ℎ−𝜏))_𝑘−𝜉(𝑡)^𝑇×𝑃₃𝜉̇(𝑡)((𝑡×𝜉𝑡+2𝛼ℎ−𝜏))_𝑘−𝜉(𝑡)^𝑇𝑃₃𝜉𝑡_𝑘−𝜉(𝑡). (30) We define the following variable 𝜙_𝜉(𝑡)=1𝜏(𝑡)^𝑡_𝑡𝑘𝜉̇(𝑠)𝑑𝑠, (31) and it is derived that 𝑒^{−2𝛼𝑡𝑡}_𝑡𝑘𝑒^2𝛼𝑠𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)𝑑𝑠⩾𝑒^−2𝛼𝑡𝑒^{2𝛼𝑡𝑘𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)=𝑒^{−2𝛼(𝑡−𝑡𝑘)𝑡}_𝑡𝑘𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠)=𝑒^{−2𝛼𝜏(𝑡)𝑡}_{(𝑡)𝑡−𝜏}𝜉̇^𝑇(𝑠)𝑃₂𝜉̇(𝑠). (32) By Jensen’s inequality, it is also shown that 𝜉̇^𝑡_{(𝑡)𝑡−𝜏}^𝑇(𝑠)𝑃₂𝜉̇(𝑠)⩽1𝑑𝑠𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝜉̇^𝑇(𝑠)𝑑𝑠×𝑃₂^𝑡_{(𝑡)𝑡−𝜏}𝜉̇(𝑠)(𝑡)1𝑑𝑠=𝜏𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝜉̇^𝑇(𝑠)𝑑𝑠×𝑃₂1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝜉̇(𝑠)(𝑡)𝜙𝑑𝑠=𝜏^𝑇_𝜉(𝑡)𝑃₂𝜙_𝜉(𝑡). (33) In particular, for 𝜏(𝑡)=0, it can be verified that the following holds: 1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝜉̇(𝑠)𝑑𝑠|_𝜏(𝑡)=0=lim_{𝜏(𝑡)→0}1𝜏(𝑡)^𝑡_{(𝑡)𝑡−𝜏}𝜉̇(𝑠)𝑑𝑠=𝜉̇(𝑡). (34) Therefore, it is shown from (30) that 𝑉̇(𝑡)+2𝛼𝑉(𝑡)⩽2𝜉̇^𝑇(𝑡)𝑃₁𝜉(𝑡)+2𝛼𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡)−𝑒^{−2𝛼𝜏(𝑡)}𝜏(𝑡)𝜙^𝑇_𝜉(𝑡)𝑃₂𝜙_𝜉(𝑡)+(ℎ−𝜏(𝑡))𝜉̇^𝑇(𝑡)𝑃₂𝜉̇(𝑡)−𝜉𝑡_𝑘−𝜉(𝑡)^𝑇𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉̇−𝜉ℎ−𝜏))^𝑇(𝑡)𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡−𝜉ℎ−𝜏))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉̇(𝑡)+2𝛼(ℎ−𝜏(𝑡))𝜉𝑡_𝑘−𝜉(𝑡)^𝑇×𝑃₃𝜉𝑡_𝑘−𝜉(𝑡). (35) We now insert free-weighting matrices by introducing the following zero terms: 2𝜉^𝑇(𝑡)𝑇^𝑇₁+𝜉̇^𝑇(𝑡)𝑇^𝑇₂×𝜉̇^𝑇(𝑡)−𝐴𝜉(𝑡)−𝐵𝐾𝜉𝑡_𝑘2𝜉=0,_𝑇(𝑡)𝑆^𝑇₁+𝜉̇^𝑇(𝑡)𝑆^𝑇₂+𝜉^𝑇𝑡_𝑘𝑆^𝑇₃×−𝜉(𝑡)+𝜉𝑡_𝑘+𝜏(𝑡)𝜙_𝜉(𝑡)=0, (36)(37) where the free-weighting matrices 𝑇₁, 𝑇₂, 𝑆₁,  𝑆₂, and 𝑆₃ are to be designed. Then we have 𝑉̇(𝑡)(𝑡)𝜉̇+2𝛼𝑉⩽2^𝑇(𝑡)𝑃₁𝜉(𝑡)+2𝛼𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡)−𝑒^{−2𝛼𝜏(𝑡)}𝜏(𝑡)𝜙^𝑇_𝜉(𝑡)𝑃₂𝜙_𝜉(𝑡)+(ℎ−𝜏(𝑡))𝜉̇^𝑇(𝑡)𝑃₂𝜉̇(𝑡)−𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡𝜉̇−𝜉ℎ−𝜏))^𝑇(𝑡)𝑃₃𝜉𝑡_𝑘(𝑡)−((𝑡−𝜉ℎ−𝜏))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇𝑃₃𝜉̇(𝑡)+2𝛼(ℎ−𝜏(𝑡))𝜉(𝑡_𝑘)−𝜉(𝑡)^𝑇×𝑃₃𝜉𝑡_𝑘(𝑡)𝜉−𝜉+2^𝑇(𝑡)𝑇^𝑇₁+𝜉̇^𝑇(𝑡)𝑇^𝑇₂×𝜉̇^𝑇(𝑡)−𝐴𝜉(𝑡)−𝐵𝐾𝜉𝑡_𝑘×2𝜉^𝑇(𝑡)𝑆^𝑇₁+𝜉̇^𝑇(𝑡)𝑆^𝑇₂+𝜉^𝑇𝑡_𝑘𝑆^𝑇₃×−𝜉(𝑡)+𝜉𝑡_𝑘+𝜏(𝑡)𝜙_𝜉(𝑡). (38) Defining the following new augmented variables 𝜉(𝑡)=𝜉^𝑇(𝑡)𝜉̇^𝑇(𝑡)𝜉^𝑇(𝑡_𝑘)𝜙^𝑇(𝑡)^𝑇,𝜉̂(𝑡)=𝜉^𝑇(𝑡)𝜉̇^𝑇(𝑡)𝜉^𝑇(𝑡_𝑘)^𝑇, (39) it is calculated that 𝑉̇(𝑡)+2𝛼𝑉(𝑡)⩽𝜉^𝑇(𝑡)Π(𝑡)𝜉(𝑡) (40) with Π(𝑡)=Π₁₁(𝑡)Π₁₂(𝑡)Π₁₃(𝑡)Π₁₄(𝑡)⋆Π₂₂(𝑡)Π₂₃(𝑡)0⋆⋆Π₃₃(𝑡)Π₃₄(𝑡)⋆⋆⋆Π₄₄(𝑡),Π₁₁(𝑡)=2𝛼𝑃₁−𝑃₃+2𝛼(ℎ−𝜏(𝑡))𝑃₃−𝑇^𝑇₁𝐴−𝐴^𝑇𝑇₁−𝑆₁−𝑆^𝑇₁,Π₁₂(𝑡)=𝑃₁+𝑇^𝑇₁−(ℎ−𝜏(𝑡))𝑃₃,Π₁₃(𝑡)=𝑃₃−2𝛼(ℎ−𝜏(𝑡))𝑃₃−𝑇^𝑇₁𝐵𝐾+𝑆^𝑇₁,Π₁₄(𝑡)=𝜏(𝑡)𝑆^𝑇₁,Π₂₂(𝑡)=(ℎ−𝜏(𝑡))𝑃₂+𝑇₂+𝑇^𝑇₂,Π₂₃(𝑡)=−(ℎ−𝜏(𝑡))𝑃₃−𝑇^𝑇₂𝐵𝐾+𝑆₂+𝑆^𝑇₂,Π₃₃(𝑡)=−𝑃₃+2𝛼(ℎ−𝜏(𝑡))𝑃₃+𝑆₃+𝑆^𝑇₃,Π₃₄(𝑡)=𝜏(𝑡)𝑆^𝑇₃,Π₄₄(𝑡)=−𝑒^{−2𝛼𝜏(𝑡)}𝜏(𝑡)𝑃₂. (41) We now consider the matrices Φ and Φ defined in (21). In fact, it can be verified that Φ is obtained from Π(𝑡)|_𝜏(𝑡)=0 with deleting the last row and last column, and Φ=Π(𝑡)|_{𝜏(𝑡)=ℎ}. Similar to the proof of Theorem 1 of [37], it can be proved that the system (19) with 𝑑(𝑡)=0 is exponentially stable.
Next, we deal with the input constraint along each axis. We define 𝑅_𝑥=100=,𝑅100100000000_𝑦=010=,𝑅010000010000_𝑧=001001=.000000001 (42) Then, the thrust along each axis can be described as 𝑢_𝑖𝑡_𝑘=𝑅_𝑖𝑢𝑡_𝑘⩽𝑢_𝑖,max,𝑖=𝑥,𝑦,𝑧. (43) Therefore, the thrust constraint (8) can be rewritten as 𝑅_𝑖𝑢𝑡_𝑘⩽𝑢_𝑖,max,𝑖=𝑥,𝑦,𝑧, (44) which is equivalent to 𝑅_𝑖𝑢𝑡_𝑘^𝑇𝑅_𝑖𝑢𝑡_𝑘⩽𝑢²_𝑖,max. (45) Furthermore, formula (45) is equal to 𝜉^𝑇𝑡_𝑘𝐾𝑅^𝑇_𝑖𝑅_𝑖𝐾𝜉𝑡_𝑘⩽𝑢²_𝑖,max. (46) Based on the previous analysis, it is known that 𝑉̇(𝑡)<0 is guaranteed if the matrix conditions (21) hold. Subsequently, 𝑉(𝑡)<𝑉(0) holds for any 𝑡>0. It is therefore reasonable to assume that there exists a scalar 𝜌>0 such that 𝑉(0)⩽𝜌. (47) Notice that 𝑉(𝑡)=𝑉₁(𝑡)+𝑉₂(𝑡)+𝑉₃(𝑡)≤𝑉(0)⩽𝜌, and 𝑉₂(𝑡)≥0,  𝑉₃(𝑡)≥0. Therefore, it is true that 𝑉(𝑡)≤𝑉₁(𝑡)<𝑉(0)⩽𝜌, (48) which implies that 𝑉₁(𝑡)<𝜌. (49) Therefore, 𝜉^𝑇(𝑡)𝑃₁𝜉(𝑡)<𝜌 holds for any 𝑡>0. In particular, let 𝑡=𝑡_𝑘; we have 𝜉^𝑇𝑡_𝑘𝑃₁𝜉𝑡_𝑘<𝜌. (50) Then, (46) can be guaranteed if (22) holds. We prove this fact. In fact, if (22) holds, we have 𝜉^𝑇𝑡_𝑘𝜌𝐾^𝑇𝑅^𝑇_𝑖𝑅_𝑖𝐾𝜉𝑡_𝑘<𝜉^𝑇𝑡_𝑘𝑢²_𝑖,max𝑃₁𝜉𝑡_𝑘. (51) By (51), we have 𝜉^𝑇𝑡_𝑘𝜌𝐾^𝑇𝑅^𝑇_𝑖𝑅_𝑖𝐾𝜉𝑡_𝑘<𝜉^𝑇𝑡_𝑘𝑢²_𝑖,max𝑃₁𝜉𝑡_𝑘<𝑢²_𝑖,max𝜌, (52) which implies that 𝜉^𝑇𝑡_𝑘𝐾^𝑇𝑅^𝑇_𝑖𝑅_𝑖𝐾𝜉𝑡_𝑘<𝑢²_𝑖,max. (53) As a result, the thrust limitation (8) can be guaranteed by (22). This completes the proof.

3.2. Disturbance Attenuation

In this subsection, we will investigate analysis of the internal stability and 𝐻_∞ performance index for the system (19). Defining a new augmented vector 𝜓(𝑡)≜[𝜉^𝑇(𝑡)𝑑^𝑇(𝑡)]^𝑇, we reconsider the system (19) with 𝑑(𝑡)≠0. We introduce the performance index 𝐽(𝑡)=^𝑡₀𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾𝑑^𝑇(𝑠)𝑑(𝑠)𝑑𝑠, (54) Then, (20) can be guaranteed by 𝐽(𝑡)<0.

Theorem 3. Consider the closed-loop system (19), given a performance index parameter 𝛾>0, let the decay rate 𝛼=0, if there exist positive definite matrices 𝑃₁, 𝑃₂, and 𝑃₃∈ℝ^9×9, and matrices 𝑇₁, 𝑇₂, 𝑆₁, 𝑆₂, and 𝑆₃∈ℝ^9×9, such that the matrix conditions (21), (22) and the following matrices conditions hold: Φ|_𝛼=0Γ₁⋆diag−𝛾²,−𝐼₉<0,Φ|_𝛼=0Γ₂⋆diag−𝛾²,−𝐼₉Γ<0,₁=−𝑇^𝑇₁𝐵^𝑇₁𝐶^𝑇₂−𝑇^𝑇₂𝐵^𝑇₁0,Γ00₂=−𝑇^𝑇₁𝐵^𝑇₁𝐶^𝑇₂−𝑇^𝑇₂𝐵^𝑇₁0.0000 (55)(56) Then the system (19) is internally stable, and the 𝐻_∞ performance index (20) is achieved.

Proof. It is well known that 𝑉̇(𝑡)<0 implies that lim_𝑡→∞𝑉(𝑡)=0. We consider a new defined function 𝑉_𝐽(𝑡)=𝑉̇(𝑡)+𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡). (57) If 𝑉_𝐽(𝑡)<0, it is shown that 𝑧^𝑇(𝑡)𝑧(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)<−𝑉̇(𝑡), (58) and it further implies that 𝐽(∞)=∫^∞₀(𝑧^𝑇(𝑠)𝑧(𝑠)−𝛾²𝑑^𝑇(𝑠)𝑑(𝑠))𝑑𝑠<−∫^∞₀𝑉̇(𝑠)𝑑𝑠=𝑉(∞)−𝑉(0)=−𝑉(0)<0. That means, in the case 𝛼=0, if 𝑉̇(𝑡)<0 and 𝑉_𝐽(𝑡)<0 hold, it can be guaranteed that the system (19) is internally stable and the 𝐻_∞ performance index (20) can be achieved successfully.
Notice that (57) can be rewritten as 𝑉̇(𝑡)+𝜉^𝑇(𝑡)𝐶^𝑇₂𝐶₂𝜉(𝑡)−𝛾²𝑑^𝑇(𝑡)𝑑(𝑡)<0. (59) Recall the zero terms (37) and notice that the term (37) with 𝑑(𝑡)≠0 becomes 2𝜉^𝑇(𝑡)𝑇^𝑇₁+𝜉̇^𝑇(𝑡)𝑇^𝑇₂×𝜉̇^𝑇(𝑡)−𝐴𝜉(𝑡)−𝐵𝐾𝜉𝑡_𝑘−𝐵₁𝑑(𝑡2𝜉)=0,_𝑇(𝑡)𝑆^𝑇₁+𝜉̇^𝑇(𝑡)𝑆^𝑇₂+𝜉^𝑇𝑡_𝑘𝑆^𝑇₃×−𝜉(𝑡)+𝜉𝑡_𝑘+𝜏(𝑡)𝜙_𝜉(𝑡)=0. (60) By similar derivation of Theorem 2, it can be proved that system (19) is internally stable if the conditions (55) hold. On the other hand, it has been proved in Theorem 2 that 𝑉̇(𝑡)<0 is guaranteed by (21)-(22). This completes the proof.

4. Simulation

In this section, simulation results will be shown to illustrate the usefulness and advantage of the proposed sampled-data control approach. We consider the numerical example in [3]: suppose that the mass of the chaser is 200 kg, that is,  𝑚=200, and the target is moving along a geosynchronous orbit with radius 𝑟=42241 km with an orbital period of 24 h. The angle velocity 𝑤=1.117×10.3 rad/s. The maximum thrust along each axis is 1000N. The disturbance vector is described as 𝑑(𝑡)=[2sin(2𝑡)2sin(2𝑡)2sin(2𝑡)]^𝑇. The upper bound of the sampling period is given as ℎ=0.5, and the decay rate is selected as 𝛼=0.02. The disturbance attenuation level is given as 𝛾=0.4. In the coordinate based on the target frame, assume that the initial state is [10,−20,10,0,0,0], and the desired holding position is (0,−1,0). Design the state feedback gain 𝐾 as𝐾=−34.38070.14900−97.38890.00510−8.70620.02540−0.1490−34.38010−0.0051−97.38800−0.0254−8.7062000−57.874500−110.000900−16.5843 (61)such that (𝐴+𝐵𝐾) is Hurwitz. Suppose that 𝑡₁=0, which means that the first sampling time instant is the initial time. The designed parameter 𝜌 is chosen as 𝜌=950. Solving the conditions (21)-(22) and (55), it is found that there exists feasible solution. Then, it can be derived that 𝑉(0)=𝑉₁(0)=𝑥(0)^𝑇𝑃₁𝑥(0)=835.9423. It is obvious the condition (47) holds, which means that our obtained solution is correct and feasible.

The simulation results are provided in Figures 1–3, which show the effectiveness of the proposed methods. The trajectories of state variable 𝑥(𝑡) with 𝑑(𝑡)=0 are shown in Figure 1; the trajectories of sampled-data control input 𝑢(𝑡) with 𝑑(𝑡)=0 are shown in Figure 2; the trajectories of Lyapunov functions 𝑉(𝑡) with 𝑑(𝑡)=0 are shown in Figure 3. It can be seen that the obtained performance is desirable.

Figure 1 

Trajectories of 𝑥(𝑡).

Figure 2 

Trajectories of 𝑢(𝑡).

Figure 3 

Lyapunov function 𝑉(𝑡).

5. Conclusion

In this paper, we have addressed the problem of sampled-data stabilization for spacecraft rendezvous with a time-dependent Lyapunov approach. A discontinuous Lyapunov functional has been constructed, the value of which does not increase in each sampling time instant. The sufficient conditions for the stability analysis and controller design are established in terms of matrix inequality technique. Future work will be focused on more complicated case; that is, the effects of network-induced phenomenon such as communication delay and packet losses are taken into account simultaneously.

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China 61104101, the Fundamental Research Funds for the Central Universities, the China Postdoctoral Science Foundation funded project (2011M500058), the Special Chinese National Postdoctoral Science Foundation under Grant 2012T50356, and the Heilongjiang Postdoctoral Fund (LBH-Z11144).