A method is presented for the selection of a set of sensor locations from a larger candidate set for the purpose of on-orbit identification and correlation of Large Space Structures. The method ranks the candidate sensor locations according to their contribution to the linear independence of the target modal partitions. In an iterative maner, the locations which do not contribute significantly are removed. The final sensor configuration tends to maximize determinant of the corresponding Fisher Information Matrix.

Sensor placement for on-orbit modal identification and correlation of large space structures

This paper presents a mathematical framework for discussion of large space structure (LSS) control theory. Within this framework, current trends in LSS control theory and related topics in general control science are surveyed.

Trends in large space structure control theory: Fondest hopes, wildest dreams

In this paper, we study the attitude control problem for spacecraft with gas jet or momentum exchange actuators, using the recent nonlinear geometric control theory. We give necessary and sufficient conditions for controllability of the system in the case that the gas jet actuators yield one, two, or three independent torques. In the case of momentum exchange devices, controllability is studied with three independent actuators, and controllability is shown to be impossible with fewer devices. The former conditions with gas jet actuators are presented in three equivalent ways, and an equivalence is established with an earlier condition by Baillieul. The local controllability problem is also studied in the case of gas jet actuators yielding two independent torques. Using these results, an algorithm stabilizing the controllable system around an equilibrium state and trajectory is outlined, as proposed by Hermes. In the situations considered, however, the linearized systems are not controllable.

Spacecraft attitude control and stabilization: Applications of geometric control theory to rigid body models

Sensor Placement for On-Orbit Modal Identification and Correlation of Large Space Structures

Preface. 1. SINGLE DEGREE OF FREEDOM SYSTEMS. Introduction. Spring-Mass System. Spring-Mass-Damper System. Forced Response. Transfer Functions and Frequency Methods. Measurement and Testing. Stability. Design and Control of Vibrations. Nonlinear Vibrations. Computing and Simulation in Matlab. Chapter Notes. References. Problems. 2. LUMPED PARAMETER MODELS. Introduction. Classifications of Systems. Feedback Control Systems. Examples. Experimental Models. Influence Methods. Nonlinear Models and Equilibrium. Chapter Notes. References. Problems. 3. MATRICES AND THE FREE RESPONSE. Introduction. Eigenvalues and Eigenvectors. Natural Frequencies and Mode Shapes. Canonical Forms. Lambda Matrices. Oscillation Results. Eigenvalue Estimates. Computational Eigenvalue Problems in Matlab. Numerical Simulation of the Time Response in Matlab. Chapter Notes. References. Problems. 4. STABILITY. Introduction. Lyapunov Stability. Conservative Systems. Systems with Damping. Semidefinite Damping . Gyroscopic Systems. Damped Gyroscopic Systems. Circulatory Systems. Asymmetric Systems. Feedback Systems. Stability in the State Space. Stability Boundaries. Chapter Notes. References. Problems. 5. FORCED RESPONSE OF LUMPED PARAMETER SYSTEMS. Introduction. Response via State Space Methods. Decoupling Conditions and Modal Analysis. Response of Systems with Damping. Bounded-Input, Bounded-Output Stability. Response Bounds. Frequency Response Methods. Numerical Simulations in Matlab. Chapter Notes. References. Problems. 6. DESIGN CONSIDERATIONS. Introduction. Isolators and Absorbers. Optimization Methods. Damping Design. Design Sensitivity and Redesign. Passive and Active Control. Design Specifications. Model Reduction. Chapter Notes. References. Problems. 7. CONTROL OF VIBRATIONS. Introduction. Controllability and Observability. Eigenstructure Assignment. Optimal Control. Observers (Estimators). Realization. Reduced-Order Modeling. Modal Control in State Space. Modal Control in Physical Space. Robustness. Positive Position Feedback Control. Matlab Commands for Control Calculations. Chapter Notes. References. Problems. 8. VIBRATION MEASUREMENT. Introduction. Measurement Hardware. Digital Signal Processing. Random Signal Analysis. Modal Data Extraction (Frequency Domain). Modal Data Extraction (Time Domain). Model Identification. Model Updating. Chapter Notes. References. Problems. 9. DISTRIBUTED PARAMETER MODELS. Introduction. Vibrations of Strings. Rods and Bars. Vibration of Beams. Membranes and Plates. Layered Materials. Viscous Damping. Chapter Notes. References. Problems. 10. FORMAL METHODS OF SOLUTION. Introduction. Boundary Value Problems and Eigenfunctions. Modal Analysis of the Free Response. Modal Analysis in Damped Systems. Transform Methods. Green's Functions. Chapter Notes. References. Problems. 11. OPERATORS AND THE FREE RESPONSE. Introduction. Hilbert Spaces. Expansion Theorems. Linear Operators. Compact Operators. Theoretical Modal Analysis. Eigenvalue Estimates. Enclosure Theorems. Oscillation Theory. Chapter Notes. References. Problems. 12. FORCED RESPONSE AND CONTROL. Introduction. Response by Modal Analysis. Modal Design Criteria. Combined Dynamical Systems. Passive Control and Design. Distribution Modal Control. Nonmodal Distributed Control. State Space Control Analysis. Chapter Notes. References. Problems. 13. APPROXIMATIONS OF DISTRIBUTED PARAMETER MODELS. Introduction. Modal Truncation. Rayleigh- Ritz-Galerkin Approximations. Finite Element Method. Substructure Analysis. Truncation in the Presence of Control. Impedance Method of Truncation and Control. Chapter Notes. References. Problems. APPENDIX A: COMMENTS ON UNITS. APPENDIX B: SUPPLEMENTARY MATHEMATICS. Index.

Vibration With Control

This paper studies the controllability and observability of the system Mq + Gq + Kq = Bu , where M is symmetric and positive-definite, G is skew-symmetric and K is symmetric. In all cases, the output equation is y = Pq + Rq . This special structure is exploited to derive relatively simple controllability and observability conditions which are shown to provide important insights on the modal behavior of the system and to furnish information on the number and positioning of sensors and actuators.

Controllability and Observability of Linear Matrix-Second-Order Systems

Reduced models and reduced controllers for systems governed by matrix-second-order differential equations are obtained by retaining those modes which make the largest contributions to quadratic control objectives. Such contributions, expressed in terms of modal data, used as mode truncation criteria, allow the statement of the specific control objectives to influence the early model reduction from very high order models which are available, for example, from finite element methods. The relative importance of damping, frequency, and eigenvector in the mode truncation decisions are made explicit for each of these control objectives: attitude control, vibration suppression and figure control. The paper also shows that using modal cost analysis (MCA) on the closed loop modes of the optimally controlled system allows the construction of reduced control policies which feedback only those closed loop modal coordinates which are most critical to the quadratic control performance criterion. In this way, the modes which should be controlled (and hence the modes which must be observable by choice of measurements), are deduced from truncations of the optimal controller.

Modal cost analysis for linear matrix-second-order systems

Modal cost analysis furnishes a promising methodology for developing dynamical models of space structures for use in control systems analysis Economy and accuracy can be attained by only retaining vibration modes that contribute significantly to an appropriately defined cost function Expressions for modal costs (especially simple for 'lightly damped' structures) are derived for attitude control, vibration suppression, and shape control These techniques are illustrated through application to a high-order finite element model of a large platform-type structure

Order reduction for models of space structures using modal cost analysis

It is a standard procedure to analyze a flexible vehicle in terms of its vibration frequencies and mode shapes. However, the entire mode shape is not needed per se, but two integrals of the mode shape, pi and hi, which correspond to the momentum and angular momentum in Mode i. Together with the natural frequencies omega-i, these modal parameters satisfy several important identities, 25 of which are derived in this paper. Expansions in terms of both constrained and unconstrained modes are considered. A simple illustrative example is included. The paper concludes with some remarks on the theoretical and practical utility of these results, and several potential extensions to the theory are suggested.

Modal Identities for Elastic Bodies, With Application to Vehicle Dynamics and Control

Current interest in extended sensing and actuation for the control of flexible spacecraft has led to the use of modern multivariable control theory and the associated concepts of controllability and observability. This paper shows how to evaluate these properties on a mode-by-mode basis for flexible spacecraft control analysis. Relatively simple criteria are derived which indicate the degree of controllability (observability) of each mode in simple literal terms. These criteria provide physical insight and practical guidance on the type, number, and positioning of sensors and actuators. The results are interpreted for force and torque actuators, and for attitude and deformation measurements. To illustrate these ideas, sample controllability and observability 'surfaces' are presented for the Purdue generic flexible spacecraft model.

P. C. Hughes

Papers

Controllability and Observability of Linear Matrix-Second-Order Systems

Modal cost analysis for linear matrix-second-order systems

Order reduction for models of space structures using modal cost analysis

Modal Identities for Elastic Bodies, With Application to Vehicle Dynamics and Control

Controllability and Observability for Flexible Spacecraft