Controllability of inherently damped large flexible space structures

Graph theoretic techniques are used to study controllability of linear systems which represent large flexible orbiting space systems with inherent damping. The controllability of the pair of matrices representing the system state and control influence matrices is assured when all states in the model are reachable in a digraph sense from at least one input and also when the term rank of a Boolean matrix whose non trivial components are based on the state and control influence matrices has a term rank of the order of the state vector. The damping matrix does not influence the required number of actuators but gives flexibility to the possibility locations of the actuators for which the system is controllable

The dynamics and control of large flexible space structures, 2. Part A: Shape and orientation control using point actuators

The equations of planar motion for a flexible beam in orbit which includes the effects of gravity gradient torques and control torques from point actuators located along the beam was developed. Two classes of theorems are applied to the linearized form of these equations to establish necessary conditions for controlability for preselected actuator configurations. The feedback gains are selected: (1) based on the decoupling of the original coordinates and to obtain proper damping, and (2) by applying the linear regulator problem to the individual model coordinates separately. The linear control laws obtained using both techniques were evaluated by numerical integration of the nonlinear system equations. Numerical examples considering pitch and various number of modes with different combination of actuator numbers and locations are presented. The independent model control concept used earlier with a discretized model of the thin beam in orbit was reviewed for the case where the number of actuators is less than the number of modes. Results indicate that although the system is controllable it is not stable about the nominal (local vertical) orientation when the control is based on modal decoupling. An alternate control law not based on modal decoupling ensures stability of all the modes

On the shape and orientation control of an orbiting shallow spherical shell structure

The dynamics of orbiting shallow flexible spherical shell structures under the influence of control actuators was studied. Control laws are developed to provide both attitude and shape control of the structure. The elastic modal frequencies for the fundamental and lower modes are closely grouped due to the effect of the shell curvature. The shell is gravity stabilized by a spring loaded dumbbell type damper attached at its apex. Control laws are developed based on the pole clustering techniques. Savings in fuel consumption can be realized by using the hybrid shell dumbbell system together with point actuators. It is indicated that instability may result by not including the orbital and first order gravity gradient effects in the plant prior to control law design

Stability analysis of large space structure control systems with delayed input

Large space structural systems, due to their inherent flexibility and low mass to area ratio, are represented by large dimensional mathematical models. For implementation of the control laws for such systems a finite amount of time is required to evaluate the control signals; and this time delay may cause instability in the closed loop control system that was previously designed without taking the input delay into consideration. The stability analysis of a simple harmonic oscillator representing the equation of a single mode as a function of delay time is treated analytically and verified numerically. The effect of inherent damping on the delay is also analyzed. The control problem with delayed input is also formulated in the discrete time domain

On the shape and orientation control of an orbiting shallow spherical shell structure

A proposed method for controlling the shape and orientation of very large shallow dish type receiver/reflectors to be used in communication, radiometry and in electronic orbital based mail systems involves connecting a rigid light weight dumbell with heavy tip masses to the shell at its apex by a spring loaded double gimballed joint with dampling. To completely damp the system transient motion in all of the important lower frequency modes, an active control system is required. A mathematical model is extended to include the effects of point actuators located at preselected positions on the shell surface. The formulation of the uncontrolled dynamics assumes an a priori knowledge of the frequencies of all the elastic modes to be incorporated within the system model. As an example, three rigid body modes and six elastic modes are included in the model and six actuators are assumed, none of which lies on a nodal line or circle

The equation of state of neutron matter, symmetry energy, and neutron star structure

We review the calculation of the equation of state of pure neutron matter using quantum Monte Carlo (QMC) methods. QMC algorithms permit the study of many-body nuclear systems using realistic two- and three-body forces in a nonperturbative framework. We present the results for the equation of state of neutron matter, and focus on the role of three-neutron forces at supranuclear density. We discuss the correlation between the symmetry energy, the neutron star radius and the symmetry energy. We also combine QMC and theoretical models of the three-nucleon interactions, and recent neutron star observations to constrain the value of the symmetry energy and its density dependence.Comment: 11 pages, 11 figure

The dynamics and control of large flexible space structures, part 7

A preliminary Eulerian formulation of the in-plane dynamics of the proposed spacecraft control laboratory experiment configuration is undertaken when the mast is treated as a cantilever type beam and the reflector as a lumped mass at the end of the beam. Frequency and mode shapes are obtained for the open loop model of the beam system and the stability of closed loop control systems is analyzed by both frequency and time domain techniques. Environmental disturbances due to solar radiation pressure are incorporated into models of controlled large flexible orbiting platforms. Thermally induced deformations of simple beam and platform type structures are modelled and expressions developed for the disturbance torques resulting from the interaction of solar radiation pressure. Noise effects in the deterministic model of the hoop/column antenna system are found to cause a degradation in system performance. Appropriate changes in the ratio of plant noise to the measurement noise and/or changes in the control weighting matrix elements can improve transient and steady state performance

The dynamics and control of large flexible space structures, 8

A development of the in plane open loop rotational equations of motion for the proposed Spacecraft Control Laboratory Experiment (SCOLE) in orbit configuration is presented based on an Eulerian formulation. The mast is considered to be a flexible beam connected to the (rigid) shuttle and the reflector. Frequencies and mode shapes are obtained for the mast vibrational appendage modes (assumed to be decoupled) for different boundary conditions based on continuum approaches and also preliminary results are obtained using a finite element representation of the mast reflector system. The linearized rotational in plane equation is characterized by periodic coefficients and open loop system stability can be examined with an application of the Floquet theorem. Numerical results are presented to illustrate the potential instability associated with actuator time delays even for delays which represent only a small fraction of the natural period of oscillation of the modes contained in the open loop model of the system. When plant and measurement noise effects are added to the previously designed deterministic model of the hoop column system, it is seen that both the system transient and steady state performance are degraded. Mission requirements can be satisfied by appropriate assignment of cost function weighting elements and changes in the ratio of plant noise to measurement noise

The dynamics and control of large flexible space structures-IV

The effects of solar radiation pressure as the main environmental disturbance torque were incorporated into the model of the rigid orbiting shallow shell and computer simulation results indicate that within the linear range the rigid modal amplitudes are excited in proportion to the area to mass ratio. The effect of higher order terms in the gravity-gradient torque expressions previously neglected was evaluated and found to be negligible for the size structures under consideration. A graph theory approach was employed for calculating the eigenvalues of a large flexible system by reducing the system (stiffness) matrix to lower ordered submatrices. The related reachability matrix and term rank concepts are used to verify controllability and can be more effective than the alternate numerical rank tests. Control laws were developed for the shape and orientation control of the orbiting flexible shallow shell and numerical results presented

The dynamics and control of large flexible space structures, 3. Part A: Shape and orientation control of a platform in orbit using point actuators

The dynamics, attitude, and shape control of a large thin flexible square platform in orbit are studied. Attitude and shape control are assumed to result from actuators placed perpendicular to the main surface and one edge and their effect on the rigid body and elastic modes is modelled to first order. The equations of motion are linearized about three different nominal orientations: (1) the platform following the local vertical with its major surface perpendicular to the orbital plane; (2) the platform following the local horizontal with its major surface normal to the local vertical; and (3) the platform following the local vertical with its major surface perpendicular to the orbit normal. The stability of the uncontrolled system is investigated analytically. Once controllability is established for a set of actuator locations, control law development is based on decoupling, pole placement, and linear optimal control theory. Frequencies and elastic modal shape functions are obtained using a finite element computer algorithm, two different approximate analytical methods, and the results of the three methods compared