Precision Attitude Stabilization: Incorporating Rise and Fall Times in Gas-Based Thrusters

T HRUSTERS have been a popular choice for microsatellite attitude control systems owing to their high power-to-weight ratio and because they use a lesser number of moving parts that can fail over time. Thruster applications to spacecraft attitude stabilization have been widely studied in the past. The reader is pointed to classical texts like Sidi [1] and Bryson [2] and the references therein for an extensive treatment of the topic. Thrusters have typically been used as on–off actuators and, therefore, are not suitable for high pointing accuracy requirements (e.g., space interferometry missions like the Terrestrial Planet Finder [3,4] and MicroArcsecond X-Ray Imaging missions [5]). There have been numerous developments in the design of variableamplitude thrustingwithout excessive loss of efficiency of operation. Fisch et al. [6] studied the variable operation of a Hall thruster without reduction in operational efficiency at low mass flow rates using segmented electrodes. Stone [7] developed a prototype variable-amplitude cold-gas thruster using a high-speed intelligent loading system that offered significant improvement in performance characteristics over traditional reaction control jets. He also stated the possibility of rescaling the system to achieve desired thrust output without significant penalties on performance. Further classical results pertaining to pulse-width pulse-frequency modulation [1] also provided efficient means of generating variable torques using on–off thrusters. These advances enabled the application of continuous feedback design for spacecraft attitude stabilization using thrusters. The feasibility of conventional thrusters to microsatellite and nanosatellite applications is, however, still limited by several factors. Microsatellite maneuvers need thrusts that are below the minimum threshold from conventional thrusters. Recent developments in micropropulsion therefore seek to reduce the lower thrust threshold and the minimum pulsewidths over which thrusters can be operated. For an extensive survey of modern micropropulsion technologies, the reader is referred to Rossi [8], Mueller [9], Stanton [10], and the references therein. Additionally, gas-based thrusters have nonzero rise and fall times that are needed to establish steady-state propellant flow (cold-gas systems) and for propellant reaction (hot-gas systems) [11]. These times vary between a fewmilliseconds to several hundred milliseconds. It is important to ensure that attitude control torque commands be implemented during the maximum thrust phase, since the thrust provided during the rise and fall phases is uncertain. Here, we consider the attitude stabilization problem of a microsatellite employing a variable-amplitude cold-gas thruster so as to ensure zero torque commands during thruster rise and fall times. To this end, an artificial control prescaling g t is defined, typically chosen to be a periodically modulated step function representing the thruster on and off schedules. The actual applied control torque to the system is g t u t , while the designed control signal is u t . Designing a continuous control law with a g t of possibly zero over severalwindows of time is a nontrivial task due to nonapplicability of standard feedback linearization results. Loria et al. [12] have cited several classes of open problems where the control is scaled by a periodically singular gain g t . One specific problem concerns stabilization of _ x f x g t; x u using a stabilizing control u for dynamics _ x f x u. To this end, Jiang et al. [13] have shown that, in general, a feedback u k x that asymptotically stabilizes _ x f x p x u may not stabilize _ x f x g t p x u with g being persistently exciting (PE). The persistence filter formulation by the authors [14] for linear single-input systems is extended to design a time-varying feedback law u t for this attitude stabilization problem. It needs to be emphasized that the persistence filter construction for the attitude stabilization problem is a significant advancement over our earlier work [14] in the sense that the filter state defined here has both state and time dependence. It is shown that this controller guarantees exponential convergence of the angular velocity and the vector part of the attitude quaternion to zero. Simulations carried out on a typical microsatellite example confirm that attitude and angular velocity errors converge to the origin for the closed-loop system. Furthermore, superimposition of the on–off window g t ensures zero commanded torques over the thruster rise and fall phases, as desired. We also compared, through simulations, the performance of the controller designed here to the one obtained from the classical proportional-derivative (PD) controller [15]. This, to the best of our knowledge, will be the first attempt at employing continuous feedback control design to compensate for nontrivial thruster rise and fall times and intermittent torque actuation. Throughout this Note, we adopt the following classical definition for the persistence of excitation and exponential stability. Definition 1 ([16], p. 72): The signal g : R ! R is said to be PE if there exist finite positive constants 1, 2, and T; such that