An optimal control approach to spacecraft rendezvous on elliptical orbit

Abstract

In this paper, we consider a linear quadratic regulator control problem for spacecraft rendezvous in an elliptical orbit. A new spacecraft rendezvous model is established. On the basis of this model, a linear quadratic regulator control problem is formulated. A parametric Lyapunov differential equation approach is used to design a state feedback controller such that the resulting closed-loop system is asymptotically stable, and the performance index is minimized. By an appropriate choice of the value of a parameter, an approximate state feedback controller is obtained from a solution to the periodic Lyapunov differential equation, where the periodic Lyapunov differential equation is solved on the basis of a new numerical algorithm. The spacecraft rendezvous mission under the controller obtained will be accomplished successfully. Several illustrative examples are provided to show the effectiveness of the proposed control design method.OPTIMAL CONTROL APPROACH TO SPACECRAFT RENDEZVOUS 159 As it is well known, the propellant storage capacity of a spacecraft rendezvous system is very small. Thus, it is important to reduce fuel consumption in the spacecraft rendezvous process. Once the chaser spacecraft runs out of fuel, the spacecraft rendezvous mission will no longer be possible to continue. Thus, the minimization of the fuel consumption during the spacecraft rendezvous mission is critically important. Consequently, the minimum fuel consumption problems in spacecraft rendezvous have been actively studied among the control community. For instance, the minimum fuel rendezvous problem is studied in [8], where the spacecraft is driven by normal power; a minimum fuel consumption for spacecraft rendezvous problem in a general central gravity force field is investigated in [9]; a minimum fuel consumption problem subject to constraint on the inverse square field is considered in [10]; low-thrust optimal rendezvous maneuver in the vicinity of an elliptical orbit is studied in [11]; differential game theory is applied in [12] to achieve the optimum lowacceleration rendezvous maneuver for spacecraft rendezvous; an exact penalty function method is used in [13] to solve a fuel optimization problem for continuous-thrust orbital rendezvous subject to collision avoidance constraint. However, the optimal control methods mentioned previously are for the design of open-loop controllers. It remains a challenge to design optimal closed-loop controllers, which are much preferred in engineering applications.In this paper, we consider a linear quadratic regulator (LQR) control problem. The nonlinear spacecraft rendezvous system is first linearized around the origin, resulting in a linear time-varying system. By considering the true anomaly as a new variable, the linear time-varying system is transformed into a linear periodic system. A parametric Lyapunov differential equation approach is proposed to design a state feedback controller such that the resulting closed-loop system is asymptotically stable, and the performance index is minimize...

An Effective Approach Control Scheme for the Tethered Space Robot System

“…In this research area, Gao et al studied an optimal control approach to spacecraft rendezvous on elliptical orbit by LQR. [28] Wang et al introduced an adaptive dynamic programming algorithm for the optimal control of the non-affine nonlinear discrete-time systems. [29] Keith Dupree used the implicit learning capabilities to learn the dynamics asymptotically and studied the optimal control of uncertain nonlinear Euler-Lagrange systems.…”

An Optimal Cooperative Control Design for State Consensus of Periodic Multiagent Systems

“…Among these, multiagent systems with linear periodic dynamics that may be employed in several applications such as satellite networks or robotic systems, are considerable. The importance of the study of periodic control systems is that the properties of periodicity may be used to achieve more suitable design, and its capability allows better describing physical dynamics with cyclic behavior such as Low Earth Orbit (LEO) satellites or robotic systems [17][18][19]. Furthermore, the theory of periodic systems provides useful tools to improve the control performance of the closed-loop systems and also is adequate enough for solving the problems of time-invariant systems where time-invariant controllers are inadequate [20,21].…”

A nearly optimal control for spacecraft rendezvous with constrained controls

“…It is not a small challenge to design optimal closed-loop controllers much preferred in engineering applications. In order to solve this problem, a parametric Lyapunov differential equation (PLDE) approach is proposed to design the state feedback controller for the spacecraft rendezvous problem, which is considered to a linear quadratic regulator (LQR) control problem such that the closed-loop system is asymptotically stable, and the performance index is minimized (Gao et al, 2014). However, input saturation is not taken into consideration, and a linear Lyapunov differential equation is required to be solved online.…”

“…Then, the spacecraft rendezvous mission under the constrained controls will be accomplished successfully. Compared with Billik (1964), Euler (1969), Stoolz andJezewski (1984), Humi (1993), Sengupta andVadali (2008), and Epenoy (2011), our approach is novel in that a nearly state feedback optimal controller is designed for the rendezvous problem; compared with Gao et al (2014), the proposed approach is novel in that constrained controls are taken into consideration, and an approximate optimal constrained state feedback controller has been tuned a priori off-line; compared with Zhou et al (2011), the proposed method is novel in that the optimization factor is introduced into the rendezvous problem in the presence of input constraints, the requirement that a linear Lyapunov differential equation is solved online can be removed, and an approximate optimal constrained state feedback controller has been tuned a priori off-line by using available data effectively. The spacecraft rendezvous requires high precision and responds well to the controls in the rendezvous process.…”