Control of structures in space

Various topics related to the control of large space structures are discussed. Equations of motion for distributed systems, eigenvalue problems, modal equations, control implementation, and the Langley beam experiment are discussed

The stability of motion of satellites with cavities partially filled with liquid

The stability and time dependent motion of a spinning satellite, simulated by a rigid body with a cavity partially filled with liquid is examined. The problem formulation, consisting of the boundary-value problem for the liquid and moment equations for the entire system is presented. Because of large Reynold's numbers involved, viscosity effects are negligible everywhere except for a thin boundary layer near the wetted surface. Using a boundary-layer analysis, the effect of the boundary layer is replaced by modified boundary conditions for the liquid. The solution of the differential equations for the inviscid problem is solved in closed form. A semi-analytical numerical solution of the inviscid equations subject to the viscous boundary condition has proved unsucessful

Control of spinning flexible spacecraft by modal synthesis

A procedure is presented for the active control of a spinning flexible spacecraft. Such a system exhibits gyroscopic effects. The design of the controller is based on modal decomposition of the gyroscopic system. This modal decoupling procedure leads to a control mechanism implemented in modular form, which represents a distinct computational advantage over the control of the coupled system. Design procedures are demonstrated for two types of control algorithms, linear and nonlinear. The first represents classical linear feedback approach, and the second represents an application of on-off control, both types made feasible by the modal decomposition scheme

Structural Information in Two-Dimensional Patterns: Entropy Convergence and Excess Entropy

We develop information-theoretic measures of spatial structure and pattern in more than one dimension. As is well known, the entropy density of a two-dimensional configuration can be efficiently and accurately estimated via a converging sequence of conditional entropies. We show that the manner in which these conditional entropies converge to their asymptotic value serves as a measure of global correlation and structure for spatial systems in any dimension. We compare and contrast entropy-convergence with mutual-information and structure-factor techniques for quantifying and detecting spatial structure.Comment: 11 pages, 5 figures, http://www.santafe.edu/projects/CompMech/papers/2dnnn.htm

Wave-based control of under-actuated flexible structures with strong external disturbing forces

Wave-based control of under-actuated, flexible systems has many advantages over other methods. It considers actuator motion as launching a mechanical wave into the flexible system which it absorbs on its return to the actuator. The launching and absorbing proceed simultaneously. This simple, intuitive idea leads to robust, generic, highly efficient, precise, adaptable controllers, allowing rapid and almost vibrationless re-positioning of the system, using only sensors collocated at the actuator-system interface. It has been very successfully applied to simple systems such as mass-spring strings, systems of Euler-Bernoulli beams, planar mass-spring arrays, and flexible three-dimensional space structures undergoing slewing motion. In common with most other approaches, this work also assumed that, during a change of position, the forces from the environment were negligible in comparison with internal forces and torques. This assumption is not always valid. Strong external forces considerably complicate the flexible control problem, especially when unknown, unexpected or unmodelled. The current work extends the wave-based strategy to systems experiencing significant external disturbing forces, whether enduring or transient. The work also provides further robustness to sensor errors. The strategy has the controller learn about the disturbances and compensate for them, yet without needing new sensors, measurements or models beyond those of standard wave-based control

Resonance modes in a 1D medium with two purely resistive boundaries: calculation methods, orthogonality and completeness

Studying the problem of wave propagation in media with resistive boundaries can be made by searching for "resonance modes" or free oscillations regimes. In the present article, a simple case is investigated, which allows one to enlighten the respective interest of different, classical methods, some of them being rather delicate. This case is the 1D propagation in a homogeneous medium having two purely resistive terminations, the calculation of the Green function being done without any approximation using three methods. The first one is the straightforward use of the closed-form solution in the frequency domain and the residue calculus. Then the method of separation of variables (space and time) leads to a solution depending on the initial conditions. The question of the orthogonality and completeness of the complex-valued resonance modes is investigated, leading to the expression of a particular scalar product. The last method is the expansion in biorthogonal modes in the frequency domain, the modes having eigenfrequencies depending on the frequency. Results of the three methods generalize or/and correct some results already existing in the literature, and exhibit the particular difficulty of the treatment of the constant mode

Interacting Growth Walk - a model for hyperquenched homopolymer glass?

We show that the compact self avoiding walk configurations, kinetically generated by the recently introduced Interacting Growth Walk (IGW) model, can be considered as members of a canonical ensemble if they are assigned random values of energy. Such a mapping is necessary for studying the thermodynamic behaviour of this system. We have presented the specific heat data for the IGW, obtained from extensive simulations on a square lattice; we observe a broad hump in the specific heat above the $\theta$-point, contrary to expectation.Comment: 4 figures; Submitted to PR

On the Exponentials of Some Structured Matrices

In this note explicit algorithms for calculating the exponentials of important structured 4 x 4 matrices are provided. These lead to closed form formulae for these exponentials. The techniques rely on one particular Clifford Algebra isomorphism and basic Lie theory. When used in conjunction with structure preserving similarities, such as Givens rotations, these techniques extend to dimensions bigger than four.Comment: 19 page

Breakdown of Conformal Invariance at Strongly Random Critical Points

We consider the breakdown of conformal and scale invariance in random systems with strongly random critical points. Extending previous results on one-dimensional systems, we provide an example of a three-dimensional system which has a strongly random critical point. The average correlation functions of this system demonstrate a breakdown of conformal invariance, while the typical correlation functions demonstrate a breakdown of scale invariance. The breakdown of conformal invariance is due to the vanishing of the correlation functions at the infinite disorder fixed point, causing the critical correlation functions to be controlled by a dangerously irrelevant operator describing the approach to the fixed point. We relate the computation of average correlation functions to a problem of persistence in the RG flow.Comment: 9 page

Surface critical behavior in fixed dimensions d<4d<4: Nonanalyticity of critical surface enhancement and massive field theory approach

The critical behavior of semi-infinite systems in fixed dimensions $d<4$ is investigated theoretically. The appropriate extension of Parisi's massive field theory approach is presented.Two-loop calculations and subsequent Pad\'e-Borel analyses of surface critical exponents of the special and ordinary phase transitions yield estimates in reasonable agreement with recent Monte Carlo results. This includes the crossover exponent $\Phi (d=3)$, for which we obtain the values $\Phi (n=1)\simeq 0.54$ and $\Phi (n=0)\simeq 0.52$, considerably lower than the previous $\epsilon$-expansion estimates.Comment: Latex with Revtex-Stylefiles, 4 page