Srikant Sukumar

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

Persistence based analysis of consensus protocols for dynamic graph networks

This article deals with the consensus problem involving agents with time-varying singularities in the dynamics or communication in undirected graph networks. Existing results provide control laws which guarantee asymptotic consensus. These results are based on the analysis of a system switching between piecewise constant and time-invariant dynamics. This work introduces a new analysis technique relying upon classical notions of persistence of excitation to study the convergence properties of the time-varying multi-agent dynamics. Since the individual edge weights pass through singularities and vary with time, the closed-loop dynamics consists of a non-autonomous linear system. Instead of simplifying to a piecewise continuous switched system as in literature, smooth variations in edge weights are allowed, albeit assuming an underlying persistence condition which characterizes sufficient inter-agent communication to reach consensus. The consensus task is converted to edge-agreement in order to study a stabilization problem to which classical persistence based results apply. The new technique allows precise computation of the rate of convergence to the consensus value.

Persistence based convergence rate analysis of consensus protocols for dynamic graph networks

Abstract This paper deals with the consensus problem of agents communicating via time-varying communication links in undirected graph networks. The highlight of the current work is to provide practically computable rates of convergence to consensus that hold for a large class of time-varying edge weights. A novel analysis technique based on classical notions of persistence of excitation and uniform complete observability is proposed. The new analysis technique for consensus laws under time-varying graphs provides explicit bounds on rate of convergence to consensus for single integrator dynamics. In the case of double integrators a minor modification to the standard relative state feedback law is shown to guarantee exponential convergence to consensus under similar assumptions as the single integrator case. The consensus problem is re-formulated in the edge agreement framework to which persistence of excitation based results apply. A novel application of results from Ramsey theory allows for proof of consensus and convergence rate estimation under switching graph topology. The current work connects classical results from nonlinear adaptive control and combinatorics to the modern theory of consensus over multi-agent networks.

Spacecraft Attitude Synchronization And Formation Keeping Using Line Of Sight Measurements

Abstract This paper considers the problem of attitude synchronization and formation keeping of two spacecrafts, with a leader. Attitudes of spacecrafts are not measured directly, instead control torques are determined from the line-of-sight (LOS) vectors between two spacecrafts, and another set of LOS vectors from a common leader. We provide distributed position and attitude control laws. Attitude control law is proposed to ensure that the two follower spacecrafts reach attitude consensus. A formation keeping position control law is also proposed so that desired distance between two follower spacecrafts and each spacecraft to the leader is achieved. Proposed position control law uses relative velocities and LOS vectors in the respective body frames of the two spacecrafts. The state feedback laws proposed in this work guarantee almost semi-global asymptotic stability of the desired closed-loop equilibrium.

Persistence Filters for Estimation: Applications to Control in Shared-Sensing Reversible Transducer Systems

We investigate state observer and feedback control design for systems with state- and time-dependent control or measurement gains. In this framework, we look at reversible transducers that are continually switched between the actuation and sensing modes at some prespecified schedule. Design and analysis of stable state-observers and feedback controllers for these classes of switched/hybrid systems are significantly complicated by the fact that, at any given instant of time, the overall system loses either controllability (during the sensing phase) or observability (during the actuation phase). In this work, we consider systems with scalar time-varying measurement gains and provide a novel observer construction that guarantees exponential reconstruction of state estimates to their true values. We go a step further to derive an exponentially stabilizing controller design that uses the state estimates resulting from our observer. This amounts to the establishment of a rather remarkable separation property of the control design. These developments hinge on a rather mild technical assumption, which can be interpreted for the reversible transducer problem as a persistent dwell time for both the sensing and actuation modes. An important feature here is that the convergence rate can be specified to any arbitrary value. Our theoretical results are validated through numerical simulations of challenging test-cases that include open-loop unstable systems. The paper also illustrates potential for nonlinear extensions of the observer based control design by considering an interesting special case.

Line-of-sight based spacecraft attitude and position tracking control

Abstract This paper considers the problem of formation control of three spacecraft consisting of one leader and two followers. The leader spacecraft controls its attitude and position to track a desired attitude and position trajectory in the Earth Centred Inertial (ECI) frame. Each follower spacecraft tracks a desired relative attitude and relative position trajectory with respect to the leader spacecraft. Absolute attitude control law for the leader and relative attitude control laws for the followers are obtained in terms of line-of-sight vectors between the spacecraft. A relative attitude determination scheme using line-of-sight vectors is also proposed. The state feedback laws proposed in this work guarantee almost global asymptotic stability of the desired closed-loop equilibrium.

Relative Attitude Trajectory Tracking Using Line Of Sight Measurements Under Spacecraft Position Dynamics

Abstract This paper considers the problem of controlling relative attitude between two spacecrafts, so as to follow a smoothly time varying desired relative attitude trajectory. The spacecrafts are modelled as rigid bodies and attitudes are represented in non-linear state space of SO(3). The attitude control law proposed does not need relative attitude measurement and is determined from line of sight (LOS) unit vectors measured by spacecrafts to each other and from each spacecraft to a common object in respective body fixed frames. Unlike existing control laws using LOS measurements, positions of spacecrafts are not considered fixed. The state feedback laws proposed in this work guarantee almost global asymptotic tracking of the desired time varying attitude trajectory, when positions of spacecrafts do not become collinear or coincide. The simulation results of the proposed control law are also shown.

Adaptive tracking of position and attitude of a underwater rigid-body

In this paper, we use the Kirchhoff equations to model of the dynamics of underwater rigid-bodies under some assumptions on the surrounding fluid. The Kirchhoff tensor, which totally characterizes the interaction of the rigid-body with the fluid is assumed to be unknown. We then propose a control law with an estimate of the Kirchhoff tensor and a corresponding update law for the estimate to track any(bounded) attitude and position trajectory asymptotically.

A jammer's perspective of reachability and LQ optimal control

This article treats two problems dealing with control of linear systems in the presence of a jammer that can sporadically turn off the control signal. The first problem treats the standard reachability problem, and the second treats the standard linear quadratic regulator problem under the above class of jamming signals. We provide necessary and sufficient conditions for optimality based on a nonsmooth Pontryagin maximum principle.

On the estimation of the consensus rate of convergence in graphs with persistent interconnections

ABSTRACT The aim of the current article is to establish myriad convergence rate estimates to consensus for time-varying graphs with persistent interaction. Several novel analysis methodologies for consensus protocols employing the notions of persistence of excitation and Lyapunov functions are provided. The estimates are compared with each other and existing literature. Numerical simulations on test examples are illustrated to support the theoretical findings.