Robust Tracking Control for Rendezvous in Near-Circular Orbits

Abstract

This paper investigates a robust guaranteed cost tracking control problem for thrust-limited spacecraft rendezvous in near-circular orbits. Relative motion model is established based on the two-body problem with noncircularity of the target orbit described as a parameter uncertainty. A guaranteed cost tracking controller with input saturation is designed via a linear matrix inequality (LMI) method, and sufficient conditions for the existence of the robust tracking controller are derived, which is more concise and less conservative compared with the previous works. Numerical examples are provided for both time-invariant and time-variant reference signals to illustrate the effectiveness of the proposed control scheme when applied to the terminal rendezvous and other astronautic missions with scheduled states signal.

1. Introduction

Autonomous rendezvous is a key operational technology in astronautic missions that involve more than one spacecraft, such as crew exchange, spacecraft assembly, maintenance, and monitoring. Generally speaking, a complete rendezvous mission can be divided into several specific phases: launch, phasing, far range rendezvous, close range rendezvous, docking, and departure [1]. Due to some similarities between far range rendezvous and close range rendezvous, researchers would collectively refer to them as the terminal rendezvous, and as the control problem is a crucial issue for the automation of spacecraft rendezvous, it has been and continues to be an appealing area of study. In this paper, we will synthesize a robust guaranteed cost tracking controller for terminal rendezvous in near-circular orbits.

Absolute navigation and relative navigation are the two major navigation methods used in a rendezvous mission. Varying with the distance between two spacecrafts, navigation methods for each phase are varied. For the launch and phasing phases where the spacecrafts are not able to detect each other, absolute navigation is in operation, while for the last four phases, as the distance between vehicles is short enough, it is common to switch to the relative navigation mode. According to the source of the navigation data, different types of controlled plants and control methods are required. The rest of this paper mainly focuses on the rendezvous problem based on the relative navigation method, and some works relying on absolute navigation are given in [2–7].

Researchers used to establish the rendezvous model on the basis of the two-body problem. Differed by the methods used in linearizing the equations of motion, rendezvous in circular, near-circular, and elliptical orbits are the three main branches that have been studied by precursors. Among these branches, rendezvous in circular and elliptical orbits seemed more attractive. Clohessy and Wiltshire [8] established the equations for satellite rendezvous in circular orbits, which is known as Clohessy-Wiltshire equations (or Hill’s equations [9]). Depending on C-W equations, Lawden put forward the primer vector theory [2], which was used by later researchers in solving the optimal 𝑁-impulse problem for rendezvous in circular orbits [10, 11]; Karr et al. [12] solved the 𝑁-impulse problem by using fuzzy control and genetic algorithms; a rendezvous controller with obstruction avoidance was realized in [13]; rendezvous under continuous thrust was considered in [14]. Although most spacecraft rendezvous missions are expected to be accomplished in circular target orbits, the eccentricities of the target orbits are not exactly equal to zero in engineering applications, which may degrade the performance of the controllers synthesized based on the hypothesis of circular target orbits. Rendezvous problems in elliptical orbits were first raised by De Vries [15] and Tschauner [16], who extended the C-W equations to be valid for elliptical target orbits, and then Carter et al. [17, 18], Zhou et al. [19], and Karlgaard and Schaub [20] gave their solutions, respectively. However, nonlinear terms were introduced in the extended C-W equations, which complicate the control tasks. The pioneer works in resolving this conflict should be attributed to Melton [21], Anthony, and Sasaki [22], who proposed the linearized dynamic models for relative motion in near-circular orbits. But as little further work has been done, it will be meaningful for us to extend their work in this paper.

Tracking control is often employed in realizing systems that are able to track reference signals. Gao and Chen [23] synthesized a network-based 𝐻_∞ output tracking controller via a linear matrix inequality (LMI) approach. In [24], a robust 𝐻_∞ PID tracking control scheme for multivariable networked control system with polyhedral uncertainties was discussed. Zhang et al. [25, 26] studied a 𝐻_∞ step tracking control problem for discrete-time nonlinear system in a networked environment with a limited capacity, and an observer-based tracking controller for discrete-time networked predictive control systems with uncertain Markov delays was designed in [27].

In order to achieve some advanced aeronautic and astronautic tasks where trajectories and velocities are required specifically, such as obstruction and detection avoidances, tracking control is often applied on the vehicles to track the preplanned reference signals. Liao et al. [28] designed a flight tracking controller by using a LMI method. Guaranteed cost control was used in the previous works to optimize the expense on fuel and the smoothness of rendezvous trajectory. Yang and Gao [29] developed a guaranteed cost output tracking controller for the terminal phase of autonomous rendezvous. Practically, thrusts produced by actuators should be bounded. A review of anterior efforts on power-limited rendezvous and a method considering both upper and lower bounds on thrust were provided by Carter and Pardis in [30]. Via a LMI approach, Yang et al. [31] designed a controller for thrust-limited rendezvous in circular orbits. However, an integrated, concise, and less conservative manner has not been proposed yet.

Motivated by the above discussions, this paper designs a guaranteed cost tracking controller for thrust-limited rendezvous in near-circular orbits via a LMI method. Based on the works of Melton [21], we put forward a relative motion model for rendezvous in near-circular orbits, where noncircularity of the target orbit is described as a parameter uncertainty. Then the rendezvous problem is formulated into a robust tracking control problem with input saturation. According to Lyapunov stabilization theory, the guaranteed cost tracking controller is cast into a convex optimization problem subject to LMI constraints, and the thrust saturation problem is solved by an improved, concise, and less conservative procedure. Finally, illustrative examples for both time-invariant and time-variant reference signals are presented to show the effectiveness of our proposed control scheme.

The remainder of this paper is organized as follows. Section 2 sets up the dynamic model and formulates the control problem. Section 3 gives the derivation of the robust tracking controller. Section 4 presents the numerical examples, and Section 5 draws the conclusion.

Notation. The notations used throughout the paper are given below. The superscript “𝑇” stands for matrix transposition; |𝑥| refers to the absolute value of 𝑥; ‖𝐱‖₂ refers to either the Euclidean vector norm or the induced matrix 2-norm. For a matrix 𝐗, sym(𝐗) stands for 𝐗+𝐗^𝑇. For a real symmetric matrix 𝐘, the notation 𝐘>𝟎 (𝐘<𝟎) is used to denote its positive- (negative-) definiteness. diag(⋯) stands for a block-diagonal matrix. In symmetric block matrices or complex matrix expressions, we use an asterisk (∗) to represent a term that is induced by symmetry. 𝐈 and 0, respectively, denote the identity matrix and zero matrix with compatible dimension. If the dimensions of matrices are not explicitly stated, they are assumed to be compatible for algebraic operation.

2. Dynamic Model and Problem Formulation

In this section, a relative motion model that describes the rendezvous process in near-circular target orbits is established based on the two-body problem. Then the rendezvous control problem is converted into a robust tracking control problem, and multiple requirements on the controller are raised.

2.1. Relative Motion Model

Suppose that a target vehicle is on a near-circular orbit with a chase vehicle adjacent. It is assumed that these two spacecrafts are only influenced by a central gravitational source. To illustrate this rendezvous system, a Cartesian coordinate system is defined with the origin fixed at the centroid of the target vehicle, the 𝑥-axis aligned with the vector 𝐫 from the Earth’s center to the origin, the 𝑧-axis aligned with the angular momentum vector of the target orbit, and the 𝑦-axis satisfying the right-hand rule, as shown in Figure 1.

Figure 1 

Cartesian coordinate system for spacecraft rendezvous.

When a chase vehicle experiences an additional perturbing force 𝐟, the equations of relative motions in above coordinates system can be derived in a method similar to that given in [32, 33]:̈𝜇𝑥=2𝑟³𝑥+𝜔̇𝑦+𝜔²̇𝑓𝑥+2𝜔𝑦+^𝑥𝑚,̈𝜇𝑦=−𝑟³𝑦−𝜔̇𝑥+𝜔²̇𝑓𝑦−2𝜔𝑥+^𝑦𝑚,̈𝜇𝑧=−𝑟³𝑓𝑧+^𝑧𝑚, (1a)(1b)(1c)where 𝑥, 𝑦, and 𝑧 are the components of the relative position; 𝜇 is the gravitational parameter; 𝜔 is the rotational rate of the 𝑥𝑦𝑧 frame relative to the gravitational source; 𝑓_𝑥, 𝑓_𝑦, and 𝑓_𝑧 are the components of the control force 𝐟; and 𝑚 is the mass of the chase vehicle.

In order to utilize linear control theory, model (1a)–(1c) should be linearized further. Generalized Lagrange’s expansion theorem [34] needed by subsequent linearization procedure is introduced as a lemma here.

Lemma 1 (see [34]). Let 𝑦 be a function of 𝑥 in terms of a parameter 𝛼 by 𝑦=𝑥+𝛼𝜙𝑦. (2) Then, for sufficiently small 𝛼, any function 𝐹(𝑦) can be expanded as a power series in 𝛼: 𝐹𝑦=𝐹(𝑥)+^∞_𝑛=1𝛼^𝑛𝑑𝑛!𝑑𝑥^{𝑛−1𝑛−1}𝜙(𝑥)^𝑛𝑑𝐹(𝑥).𝑑𝑥 (3)

According to the conversions between the orbit parameters and Kepler’s time equation𝐸=𝑀+𝑒sin𝐸, nonlinear terms in (1a)–(1c) can be rewritten as the functions of 𝐸, where 𝐸 denotes the eccentric anomaly of the target vehicle, 𝑀=𝑛(𝑡−𝑡_𝑝) is the mean anomaly of the target vehicle, 𝑡_𝑝 is the time of periapsis passage, and 𝑒 is the eccentricity of the target orbit. When eccentricity 𝑒 is sufficiently small, by Lemma 1, the nonlinear terms can be expanded as power series in constant 𝑒. Truncated at order 𝑒, we obtain the linearized results in (4a)–(4d):𝜇𝑟³=𝑛²𝑎𝑟³=𝑛²11−𝑒cos𝐸³≈𝑛²(),ℎ1+3𝑒cos𝑀𝜔=𝑟²1=𝑛1−𝑒cos𝐸²(),𝜔≈𝑛1+2𝑒cos𝑀₂=ℎ𝑟²²=𝑛²11−𝑒cos𝐸⁴≈𝑛²(),̇1+4𝑒cos𝑀𝜔=−2ℎ𝑟³=−2𝑛²𝑒sin𝐸(1−𝑒cos𝐸)⁴≈−2𝑒𝑛²sin𝑀, (4a)(4b)(4c)(4d)where𝑛=𝜇/𝑎³ is the mean motion of the target vehicle; ℎ is the angular momentum of the target orbit; 𝑟 is the radius of the target vehicle; and 𝑎 is the semimajor axis of the target orbit. By forming state vector𝐱(𝑡)=[𝑥,𝑦,𝑧,𝑥̇,𝑦,̇𝑧]̇^𝑇 and input vector𝐮(𝑡)=[𝑓_𝑥,𝑓_𝑦,𝑓_𝑧]^𝑇 and substituting (4a)–(4d) into (1a)–(1c), the first-order relative motion model for rendezvous in near-circular orbits can be rewritten in a matrix form as 𝐱̇(𝑡)=(𝐀+Δ𝐀)𝐱(𝑡)+𝐁𝐮(𝑡), (5) where𝐀=0001000000100000013𝑛²0002𝑛0000−2𝑛0000−𝑛²1000,𝐁=𝑚,000000000100010001Δ𝐀=00000000000000000010𝑒𝑛²cos𝑀−2𝑒𝑛²sin𝑀004𝑒𝑛cos𝑀02𝑒𝑛²sin𝑀𝑒𝑛²cos𝑀0−4𝑒𝑛cos𝑀0000−3𝑒𝑛²,cos𝑀000 (6)where norm-bounded matrix Δ𝐀 that contains the eccentricity of the target orbit is defined as the noncircular uncertainty. Moreover, the uncertain matrix Δ𝐀 can be factorized as Δ𝐀=𝐄₁Λ𝐄₂, (7) where Λ bounded by Λ^𝑇Λ<𝐈 is a time-variant matrix with Lebesgue measurable elements; and 𝐄₁ and 𝐄₂ are two constant matrices with proper dimensions.

Adopting (5) as the plant model for rendezvous system, the controller can be simpler and more robust. As mentioned in Section 1, most of the rendezvous missions were conducted in near-circular orbits. So for engineering applications, the wisdom of synthesizing the rendezvous controllers on the basis of model (5) is that, comparing with the circular-orbit model, the near-circular-orbit model (5) is more precise, which can guarantee the robustness and performance of the controllers, and that, comparing with the control schemes based on the elliptical-orbit model, the controllers based on model (5) are easier to design.

2.2. Problem Formulation

Assume that reference state vector 𝐱_𝑟, which is expected to be tracked by the chase vehicle, is generated in real time or has been scheduled beforehand and can be either time invariant or time variant. To assess the error between state vector 𝐱(𝑡) and the reference signal 𝐱_𝑟, tracking error 𝐱_𝑒(𝑡) is defined as 𝐱_𝑒(𝑡)=𝐱(𝑡)−𝐱_𝑟. (8) Consider the following state feedback control law: 𝐮(𝑡)=−𝐊𝐱_𝑒(𝑡), (9) where 𝐊 is the state feedback gain matrix to be determined. In order to evaluate the performance of the tracking controller, a quadratic cost function at time 𝜏(≥0) is defined as 𝐽(𝜏)=^∞_𝜏𝐱^𝑇_𝑒(𝑡)𝐐𝐱_𝑒(𝑡)+𝐮^𝑇(𝑡)𝐑𝐮(𝑡)𝑑𝑡, (10) where 𝐐, a positive symmetric matrix, is the state weighting matrix related to the smoothness of the rendezvous trajectory and convergence rate of the tracking error; 𝐑, a positive symmetric matrix, is the control weighting matrix related to the fuel cost of the chase vehicle. Thrust constraints are also considered in our research, which can be formulated as 𝑓_𝑖=𝑢_𝑖(𝑡)≤𝑢_𝑖,max𝑖=𝑥,𝑦,𝑧, (11) where 𝑢_𝑖(𝑡) is the control thrust along the 𝑖-axis and 𝑢_𝑖,max denotes the maximum control thrust that propellers can generate along the 𝑖-axis. For the purpose of dividing the input vector 𝐮(𝑡) into 𝑢_𝑥(𝑡), 𝑢_𝑦(𝑡), and 𝑢_𝑧(𝑡), which is helpful in designing the input saturation control law, matrices 𝐔_𝑖(𝑖=𝑥,𝑦,𝑧) are introduced. By defining 𝐔_𝑥=[1,0,0]^𝑇[1,0,0], 𝐔_𝑦=[0,1,0]^𝑇[0,1,0], and 𝐔_𝑧=[0,0,1]^𝑇[0,0,1], (11) can be rewritten as 𝐔_𝑖𝐮(𝑡)≤𝑢_𝑖,max𝑖=𝑥,𝑦,𝑧. (12)

According to relative motion model (5) and restrictions on the controller described above, the rendezvous control problems to be studied can be stated as follows.

Based on controlled plant (5), design a robust tracking controller that meets the following requirements.(i)The tracking error 𝐱_𝑒 converges to 0 or lim_𝑡→∞𝐱(𝑡)=𝐱_𝑟; in short, the chase vehicle can track a preplanned trajectory in a preplanned velocity.(ii)The control inputs along each axis 𝑢_𝑖(𝑡) should not exceed the given upper bounds 𝑢_𝑖,max in (11).(iii)During the rendezvous, the quadratic cost function defined in (10) should be the minimum, which means that the smoothness of the trajectory, the convergence rate of the tracking error, and the cost in fuel should be the optimal during the rendezvous.

3. Main Results

In this section, a robust guaranteed cost tracking controller with input saturation for autonomous rendezvous is designed via a LMI method, and a convex optimization problem for solving the controller is presented at the end of this section.

First of all, a lemma needed by the subsequent derivation is given. Proofs and applications for this lemma can be found in [35].

Lemma 2 (see [35]). Given matrices 𝐘=𝐘^𝑇, 𝐃, and 𝐄 of appropriate dimensions, 𝐘+𝐃𝐅𝐄+𝐄^𝑇𝐅^𝑇𝐃^𝑇<𝟎, (13) for all 𝐅 satisfying 𝐅^𝑇𝐅≤𝐈, if and only if there exists a scalar 𝜀>0 such that 𝐘+𝜀𝐃𝐃^𝑇+𝜀⁻¹𝐄^𝑇𝐄<𝟎. (14)

For each moment, the reference signal 𝐱_𝑟 can be treated as a constant vector. Substituting (8) and (9) into (5), the closed-loop error dynamic model can be represented as 𝐱̇_𝑒(𝑡)=𝐱̇(𝑡)=(𝐀+Δ𝐀−𝐁𝐊)𝐱_𝑒(𝑡)+(𝐀+Δ𝐀)𝐱_𝑟, (15) where the first term on the right-hand side plays a role in eliminating the tracking error or driving state vector 𝐱(𝑡) to approach reference signal 𝐱_𝑟 and the second term contributes to the tracking system by maintaining the state vector around the reference signal. To track the reference state signal, the designers should determine the state feedback gain matrix 𝐊 to ensure the closed-loop error system’s (15) asymptotical stability at the point 𝐱_𝑒(𝑡)=𝟎. As the reference signal has little influence on the self-stability of the error system, it is reasonable to assume it to be 0 temporarily, and then the equation of the closed-loop error system can be expressed as 𝐱̇_𝑒(𝑡)=(𝐀+Δ𝐀−𝐁𝐊)𝐱_𝑒(𝑡). (16) Equation (16) transforms the rendezvous tracking control problem into a stabilization problem. The following theorem gives the sufficient conditions for the existence of our proposed robust tracking controller.

Theorem 3. Consider the closed-loop system (16) with the state feedback control law in (9). For a given maximum tolerant tracking error 𝐱_𝑒,max, if there exist a positive symmetric matrix 𝐗, a matrix 𝐘 with proper dimensions, and positive scalars 𝜀 and 𝜌 satisfying Ψ𝐗𝐄^𝑇₂𝐘^𝑇𝐗∗diag−𝜀𝐈,−𝐑⁻¹,−𝐐⁻¹<𝟎,−𝜌⁻¹𝐈𝐔_𝑖𝐘∗−𝑢²_𝑖,max𝐗<𝟎,−𝜌⁻¹𝜌⁻¹𝐱^𝑇_𝑒,max∗−𝐗<𝟎, (17)(18)(19) where Ψ=sym(𝐀𝐗−𝐁𝐘)+𝜀𝐄₁𝐄^𝑇₁, (20) then there exists a proper controller such that the closed-loop system (16) is asymptotically stable at 𝐱_𝑒(𝑡)=𝟎, quadratic cost (10) has an upper bound 𝜌, and control forces along the 𝑖-axis (𝑖=𝑥,𝑦,𝑧) are constrained below the maximum control thrust 𝑢_𝑖,max.

Proof. Consider the Lyapunov function𝑉(𝑡)=𝐱^𝑇_𝑒(𝑡)𝐏𝐱_𝑒(𝑡), where 𝐏 is a positive symmetric matrix. Substituting (16) into the derivative of 𝑉(𝑡), we have 𝑉̇(𝑡)=sym𝐱^𝑇_𝑒(𝑡)𝐏(𝐀+Δ𝐀−𝐁𝐊)𝐱_𝑒(𝑡). (21) As is well known, the asymptotic stability of (16) can be ensured by 𝑉̇(𝑡)<0.
When matrices 𝐏, 𝐊, 𝐐, and 𝐑 are assigned appropriately, the following inequalities hold: 𝑉̇(𝑡)≤−𝐱^𝑇_𝑒(𝑡)𝐐𝐱_𝑒(𝑡)+𝐮^𝑇(𝑡)𝐑𝐮(𝑡)<0. (22) Integrating (22) from 𝜏(≥0) to ∞ and noticing that 𝐱_𝑒(𝑡)→𝟎 as 𝑡→∞, we obtain 𝐽(𝜏)≤𝑉(𝜏)=𝐱^𝑇_𝑒(𝜏)𝐏𝐱_𝑒(𝜏)=𝐽_max(𝜏), (23) where 𝑉(𝜏) also denoted as 𝐽_max(𝜏) is defined as the upper bound of the quadratic cost at time 𝜏. But as the quadratic cost 𝐽(𝜏) and its upper bound 𝐽_max(𝜏) in (23) are acquired when reference signal 𝐱_𝑟 is assumed to be 0, these two parameters can only describe the system’s performance in reducing the tracking error, instead of the cost in maintaining the chase vehicle at a predetermined position in a predetermined velocity when reference signal is nonzero. And as the quadratic cost in maintaining is determined by the reference signal and the time for holding, it is boundless when terminal time is unknown and may not be distinguished by different controllers. So it will be meaningless, if we take this consumption into consideration.
Consider the situation where reference signal 𝐱_𝑟 is time variant. The upper bound 𝐽_max may change as the reference signal changes, which makes the guaranteed cost controller difficult to design. To find a simple and uniform method to solve the problem, maximum tolerant tracking error 𝐱_𝑒,max is defined. Ensuring that (23) hold during rendezvous is one of the basic requirements for the subsequent derivation. So when reference signal is time invariant, one of a feasible assignment for 𝐱_𝑒,max is 𝐱_𝑒(0); for the time-variant reference signal, 𝐱_𝑒,max should be assigned to the maximum tracking error that the users can tolerate and meets ‖𝐱_𝑒,max‖₂≥max‖𝐱_𝑒(𝑡)‖₂. Then the upper bound of the quadratic cost 𝐽_max can be expressed uniformly as 𝐽_max=𝐱^𝑇_𝑒,max𝐏𝐱_𝑒,max. (24)
Substituting (9) and (21) into inequalities (22), the inequalities can be transformed into sym[𝐏(𝐀+Δ𝐀−𝐁𝐊)]+𝐐+𝐊^𝑇𝐑𝐊<𝟎. (25) When inequality (25) holds, it means that the error system is asymptotically stable at 𝐱_𝑒(𝑡)=𝟎 and (24) is an upper bound of the quadratic cost. To avoid solving inequality (25) that contains time-variant term Δ𝐀, substituting (7) into (25), there is sym[𝐏(𝐀−𝐁𝐊)]+𝐐+𝐊^𝑇+𝐑𝐊𝐏𝐄₁Λ𝐄₂+𝐄^𝑇₂Λ^𝑇𝐏𝐄₁^𝑇<𝟎. (26) By Lemma 2, there exists a positive scalar 𝜀 that can ensure (26) by 𝛀+𝜀𝐏𝐄₁𝐏𝐄₁^𝑇+𝜀⁻¹𝐄^𝑇₂𝐄₂<𝟎, (27) where 𝛀=sym[𝐏(𝐀−𝐁𝐊)]+𝐐+𝐊^𝑇𝐑𝐊. (28) By Schur complement, inequality (27) is equivalent to Ξ𝐄^𝑇₂𝐊^𝑇𝐈∗diag−𝜀𝐈,−𝐑⁻¹,−𝐐⁻¹<𝟎, (29) where Ξ=sym[𝐏(𝐀−𝐁𝐊)]+𝜀𝐏𝐄₁𝐄^𝑇₁𝐏^𝑇. (30) Let 𝐗=𝐏⁻¹ and 𝐘=𝐊𝐏⁻¹. Pre- and postmultiplying (29) by diag(𝐗,𝐈), inequality (29) can be transformed into LMI (17). To minimize the upper bound 𝐽_max, assume that there exists a scalar 𝜌 that meets 𝐽_max=𝐱^𝑇_𝑒,max𝐗⁻¹𝐱_𝑒,max<𝜌. (31) By Schur complement, inequality (31) is equal to −𝜌𝐱^𝑇_𝑒,max∗−𝐗<𝟎. (32)
In order to design a controller with input saturation, squaring both sides of (12) and dividing the result by 𝑢²_𝑖,max, we can readily have 𝑢⁻²_𝑖,max𝐔_𝑖𝐊𝐱_𝑒(𝑡)^𝑇𝐔_𝑖𝐊𝐱_𝑒(𝑡)≤1. (33) According to (31) and the assignment rules of the maximum tolerant tracking error 𝐱_𝑒,max, inequalities (34) hold 𝜌⁻¹𝐱^𝑇_𝑒(𝑡)𝐏𝐱_𝑒(𝑡)<𝜌⁻¹𝐱^𝑇_𝑒,max𝐏𝐱_𝑒,max<1. (34) Combing (33) and (34), thrusts can be constrained below 𝑢_𝑖,max, when inequalities (35) hold 𝑢⁻²_𝑖,max𝐔_𝑖𝐊^𝑇𝐔_𝑖𝐊<𝜌⁻¹𝐏. (35) From inequalities (34), we can see that 𝐱_𝑒,max should be assigned carefully by considering both the mobility of the chase vehicle and the tendency of the reference signal, or if the situation, 𝐱_𝑒(𝑡)>𝐱_𝑒,max, happens, inequality (35) will be invalid, which may result in thrust exceeding, performance degradation, or even system instability.
Based on Schur complements, inequalities (35) can be ensured by −𝜌⁻¹𝐈𝐔_𝑖𝐊∗−𝑢²_𝑖,max𝐏<𝟎, (36) where 𝑖=𝑥,𝑦,𝑧. Pre- and postmultiplying (36) by diag(𝐈,𝐏⁻¹) and by the definitions, 𝐗=𝐏⁻¹ and 𝐘=𝐊𝐏⁻¹, matrix inequalities (36) are equivalent to (18). When 𝜌⁻¹ is treated as a variable, to linearize matrix inequality (32), pre- and postmultiplying it by diag(𝜌⁻¹,𝐈) yield LMI (19). This completes the proof.

From (31) it is clear to see that the upper bound of quadratic cost 𝐽_max will be minimized, if we take action to minimize the positive scalar 𝜌. So to minimize 𝜌, another positive scalar 𝑤 is introduced, which meets 𝑤>𝜌>0. By Schur complement, the optimization problem of 𝐽_max and 𝜌 is converted to an optimization problem of 𝑤: −𝑤11−𝜌⁻¹<0. (37) Finally, the guaranteed cost controller with input saturation for autonomous rendezvous in near-circular orbits can be obtained by solving the following convex optimization problem: min_{𝜀,𝜌−1,𝐗,𝐘}(),(),()().𝑤,s.t.171819and37 (38) Problem (38) can be solved by the commercial software. An optimal solution that consists of 𝜀, 𝜌, 𝑤, 𝐗, and 𝐘 will be obtained, and the state feedback gain matrix can be figured out by 𝐊=𝐘𝐗⁻¹.

In [36–41], control thrusts are limited by a preset upper bound on the performance cost, which inevitably causes the systems to be conservative. However, we improve the previous works in inequality (35) by restraining the control forces with an optimal and minimum upper bound 𝜌 that is being optimized while solving (38), so, from Theorem 3 and optimization problem (38), a less conservative robust tracking controller with input saturation can be obtained.

4. Simulation Results and Discussion

In this section, two examples are presented to demonstrate the effectiveness and advantages of the control scheme presented above. The first example is for the situation in which the reference signal is time invariant, while the other example is for the situation where the reference signal is time variant, and both of the simulations are carried out in a two-body system.

Firstly, a simulated scene of spacecraft rendezvous is set. Consider a pair of adjacent spacecrafts. The target vehicle is circling on a low earth orbit with semimajor axis 𝑎=7082.253 km and eccentricity 𝑒=0.05. Thus, the mean motion of the target vehicle is 𝑛=1.06×10⁻³ rad/s. The mass of the chase vehicle is 200 kg. The maximum control thrusts along the 𝑥-, 𝑦-, and 𝑧-axis are 50 N, 50 N, and 20 N.

Then, in both examples, the matrices in (7) are assigned as 𝐄₁=,𝐄00000000000000000002𝑒4𝑒08𝑒02𝑒004𝑒0000006𝑒0₂=𝑛²000000𝑛²00002.5𝑛²0𝑛²0𝑛000.25𝑛²0−𝑛0000𝑛²00000000𝑛²,Λ=diag(sin𝑀,−sin𝑀,cos𝑀,cos𝑀,−0.5cos𝑀,cos𝑀), (39)where the mean anomaly 𝑀=𝑛𝑡. To figure out the controller in the form of (9), we should acquire the state feedback gain matrix 𝐊 by solving convex problem (38), such that closed-loop system (16) is asymptotically stable at 𝐱_𝑒(𝑡)=𝟎 with an upper bound 𝐽_max of quadratic cost and input saturation. As mentioned in the introduction, the propulsive thrust discussed in these two examples can vary continuously.

4.1. Example for Time-Invariant Reference Signal

Suppose that the chase vehicle starts at a point which is 3000 m, −4000 m, and 20 m from the target vehicle along the 𝑥-,   𝑦-, and 𝑧-axis and the initial velocities of the chase vehicle are −3 m/s, 4 m/s, and −0.02 m/s along each axis. Thus, the initial state vector of the chaser is 𝐱(0)=[3000,−4000,20,−3,4,−0.02]^𝑇. In this example, although only the case where 𝐱_𝑟=𝟎 is discussed, the subsequent procedures are also valid for the cases where reference signals are nonzero constants. The practical situation for this example can be simply described as follows: a chase vehicle approaches a target vehicle via optimal trajectory and strategy, which is a basic problem in spacecraft rendezvous. So rendezvous accomplished when state vector equals 0. Based on the guidelines for assigning the maximum tolerant tracking error, in this case, 𝐱_𝑒,max should be assigned to be the same as 𝐱(0). Solving convex optimization problem (38), we obtain (for brevity, only parts of the result are listed)𝐗=^{0.0097−0.00161.3846×10−4−4.8897×10−59.8620×10−6−1.9429×10−6−0.00160.00997.3825×10−52.3275×10−5−5.0499×10−5−5.9134×10−71.3846×10−47.3825×10−50.0099−7.8933×10−7−4.2684×10−7−4.8645×10−5}_{−4.8897×10−52.3275×10−5−7.8933×10−77.1142×10−7−3.9821×10−71.8506×10−89.8620×10−6−5.0499×10−5−4.2684×10−7−3.9821×10−76.4743×10−7−5.8225×10−11−1.9429×10−6−5.9134×10−7−4.8645×10−51.8506×10−8−5.8225×10−114.4400×10−7},𝐘=^{4.6785×10−5−3.7093×10−56.1094×10−71.8544×10−7−1.2199×10−71.3959×10−9}_{−2.0179×10}⁻⁵_1.3633×10⁻⁵_{−4.6829×10}⁻⁷_{−2.7519×10}⁻⁷_4.7088×10⁻⁷_{−2.4866×10}⁻¹⁰_{−1.4572×10−6−1.0233×10−6−9.4717×10−69.9580×10−8−4.4571×10−92.7637×10−7}. (40)Therefore, by assumptions, 𝐗=𝐏⁻¹ and 𝐘=𝐊𝐏⁻¹, the state feedback gain matrix is𝐊=𝐘𝐗⁻¹=0.0090−0.00534.7352×10⁻⁵0.9754−0.13683.5442×10⁻⁵−0.00230.0081−2.0080×10⁻⁵−0.04951.36509.9137×10⁻⁷0.00154.8836×10⁻⁴0.00460.31850.20751.1150. (41)

With the time-invariant reference signal 𝐱_𝑟 and controller solved above, the relative distance between two spacecrafts during rendezvous is depicted in Figure 2. To check if the control forces acting on the chase vehicle meet requirements (ii), the control forces along each axis during the rendezvous are presented in Figure 3 and Figure 4 shows the rendezvous trajectory of the chase vehicle during the first 3500 s.

(a)

(b)

Figure 2 

Relative distances between two spacecrafts along each axis.

(a)

(b)

Figure 3 

Propulsive thrusts of the chase vehicle along each axis.

Figure 4 

Rendezvous trajectory of the chase vehicle.

From Figures 2 and 4, it can be seen that the tracking errors or relative distances along each axis converge to zero asymptotically, though some fluctuations exist along the 𝑧-axis initially, implying that, in spite of the parameter uncertainty, the controller can stabilize the system effectively with a smooth rendezvous trajectory and low fuel cost. In Figure 3, the control forces along each axis are all below the maximum control forces, indicating that the input constraints are well guaranteed by our controller. Moreover, the largest input force of three axes, −44.6226 N generated by the propulsive thruster along the 𝑥-axis at the beginning of the rendezvous, is slightly less than 50 N showing that our method is less conservative than the previous works in restraining the thrust magnitudes.

4.2. Example for Time-Variant Reference Signal

In order to study the controller’s performance in tracking a time-variant reference signal, a straight-line rendezvous trajectory is planned in advance. The initial state vector is set to 𝐱(0)=[−7000,0,0,0,0,0], which means that the chase vehicle starts at the position that is just below the target vehicle with no initial relative velocity. Step, ramp, and acceleration signals are applied to be the reference velocity signal along the 𝑥-axis, and the reference velocities along the 𝑦- and 𝑧-axis are assigned to zeros, which are given below 𝑥̇_𝑟=−0.000024𝑡²()+0.024𝑡,0<𝑡≤500s,6,500s<𝑡≤1000s,6−0.012𝑡−1000,1000s<𝑡≤1500s,0,1500s<𝑡≤2000s. (42) From (42), we can find that the largest required acceleration along the 𝑥-axis is 0.024 m/s², which is less than 0.25 m/s², the maximum acceleration that can be generated by the chase vehicle. Referring to the initial state 𝐱(0) and integrating the reference velocity given in (42), we obtain the reference position signal along the 𝑥-axis as 𝑥_𝑟=−0.000008𝑡³+0.012𝑡²−7000,0<𝑡≤500s,6𝑡−8000,500s<𝑡≤1000s,−0.006𝑡²+18𝑡−14000,1000s<𝑡≤1500s,−500,1500s<𝑡≤2000s. (43) As it is designed, the chase vehicle is in the accelerated motion during the first 500 s, uniform motion during the second 500 s, uniformly decelerated motion in the third 500 s, and position holding in the last 500 s.

To demonstrate the maximum tolerant tracking error’s impacts on the rendezvous system, two groups of parameters are adopted and compared. For the precise group, we set the tolerant error to 𝐱_𝑒,max=[5,1,1,0.5,0.1,0.1]^𝑇. For the rough group, the tolerant error is set to 𝐱_𝑒,max=[50,1,1,5,0.1,0.1]^𝑇. Solving convex problem (38), for brevity, only parts of the precise group’s result are listed𝐗=^{0.09843.8317×10−55.3660×10−4−0.0049−1.0531×10−4−1.1207×10−53.8317×10−50.0985−1.2901×10−51.0183×10−4−0.00494.4791×10−65.3660×10−4−1.2901×10−50.0967−9.7569×10−6−4.3481×10−6−0.0049}_{−0.00491.0183×10−4−9.7569×10−64.9120×10−41.4237×10−72.4105×10−6−1.0531×10−4−0.0049−4.3481×10−61.4237×10−74.9256×10−4−3.1390×10−7−1.1207×10−54.4791×10−6−0.00492.4105×10−6−3.1390×10−74.8237×10−4},𝐘=7.0369×10⁻⁵−1.8340×10⁻⁵−3.4114×10⁻⁴0.00501.1126×10⁻⁶5.2814×10⁻⁵6.3564×10⁻⁶−6.3503×10⁻⁷−3.1493×10⁻⁵−3.1379×10⁻⁷0.00501.5873×10⁻⁶2.5518×10⁻⁴7.4186×10⁻⁶−0.00138.4057×10⁻⁶−3.6874×10⁻⁶.0.0050 (44)By assumptions, 𝐗=𝐏⁻¹ and 𝐘=𝐊𝐏⁻¹, the state feedback gain matrix of the precise group is𝐊=𝐘𝐗⁻¹=1.0046−0.0216−0.011220.1737−0.0042−0.08130.02091.00020.0020−0.005420.10760.0281−0.00781.5830×10⁻⁴1.0438−0.14220.015020.8789. (45)

With reference signals in (42), (43), and the controller solved above, the relative distance and velocity between two spacecrafts during rendezvous are illustrated in Figure 5. According to relative motion model (5), the motions along the 𝑥- and 𝑦-axis are coupling. So Figure 6 presents the control forces along these two axes of the precise group.

(a)

(b)

Figure 5 

Relative position and velocity between two spacecrafts along the 𝑥-axis.

Figure 6 

Propulsive thrusts of the chase vehicle along the 𝑥- and 𝑦-axis.

Figure 5 illustrates that the controller can track the step, ramp, and acceleration signals well and the maximum tolerant tracking error 𝐱_𝑟,max does have impacts on the performance of the controller. Generally speaking, when 𝐱_𝑟,max is compatible with the mobility of the chase vehicle and the tendency of the reference signal, the smaller the error is, the more precise the tracking system will be. From Figure 6, the largest control thrust of the precise group is 12.2830 N smaller than 50 N, and we can conclude that, if 𝐱_𝑟,max is assigned properly, the input constraint can be well guaranteed by the controller even in the situation where reference signal is time variant.

5. Conclusions

This paper has discussed a guaranteed cost tracking control problem for spacecraft rendezvous with an upper bound on thrust. A relative motion model with parameter uncertainty for rendezvous in near-circular orbits has been established. Via a LMI approach, an integrated, concise, and less conservative control scheme has been proposed. Then the controller has been demonstrated by two numerical examples with time-invariant and time-variant reference signals. The results show that our control scheme is effective for the terminal phase of rendezvous with all purposed requirements met, and, due to its strong ability in tracking reference signal, this control method should also be valid for flying by, departure, and other potential missions with planned path.

Conflict of Interests

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

Acknowledgments

This work is partially supported by the National Natural Science Foundation of China (Grant no. 61104101), the China Postdoctoral Science Foundation Funded Project (Grant no. 2011M500058), the Special Chinese National Postdoctoral Science Foundation (Grant no. 2012T50356), the Heilongjiang Postdoctoral Fund (Grant no. LBH-Z11144), the open fund of national defense key discipline laboratory of Micro-Spacecraft Technology (Grant no. HIT.KLOF.MST.2012006), and Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20112302120011), and we wish to thank Robert G. Melton at Pennsylvania State University for his suggestion on this paper.