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Publications and source records attributed to Meirovitch, L..
Work during the period January 1 to June 30, 1985 has concentrated on the completion of the derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE) as well on the development of a control scheme for the maneuvering of the spacecraft. The report consists of a paper presented at the Fifth Symposium on Dynamics and Control of Large Structures, June 12 to 14, 1985 at Blacksburg, VA.
This paper is concerned with the derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE). For future reference, the equations of motion of a similar structure orbiting the earth are also derived. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is presented where the quantities defining the rigid-body maneuver are assumed to be relatively large, with the elastic deformations and deviations from the rigid-body maneuver being relatively small. The perturbation equations have the form of linear, non-self-adjoint equations with time-dependent coefficients. An active control technique can then be formulated to permit maneuvering of the spacecraft and simultaneously suppressing the elastic vibration.
Work during the period July 1 - December 31, 1985, has concentrated on the application of the equations derived in the preceding period to the maneuvering and vibration suppression of the Spacecraft Control Laboratory Experiment (SCOLE) model. Two different situations have been considered: (1) a space environment and (2) a laboratory environment. This report covers the first case and consists of a paper entitled Maneuvering and Vibration Control of Flexible Spacecraft, presented at the Workshop on Structural Dynamics and Control Interaction of Flexible Structures, Marshall Space flight Center, Huntsville, AL, April 22 to 24, 1986. The second case will be covered in the report for the next period.
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The problem of control of a large-order flexible structure in the form of a plate-like lattice by the Independent Modal-Space Control (IMSC) method is presented. The equations of motion are first transformed to the modal space, thus obtaining internal (plant) decoupling of the system. Then, the control laws are designed in the modal space for each mode separately, so that the modal equations of motion are rendered externally (controller) decoupled. This complete decoupling applies both to rigid-body modes and elastic modes. The application of linear optimal control, in conjunction with a quadratic performance index, is first reviewed. A solution for high-order systems is proposed here by the IMSC method, whereby the problem is reduced to a number of modal minimum-fuel problems for the controlled modes.
The most common technique for the control of structures is modal control. In modal control, the differential equations in terms of actual coordinates are replaced by a set of ordinary differential equations in terms of the modal coordinates known as modal equations. In designing feedback controls in conjunction with the modal equations, one must know the modal states for the modes targeted for control. The sensors measure actual states, however. The modal states can be estimated by means of a Luenberger observer or modal filters. The modal filters produce estimates of the modal states from distributed measurements of the states. If distributed measurements are not available, then they can be reconstructed from measurements at discrete points via interpolation. This paper examines various questions associated with the implementation of modal filters, such as the effect of choice of interpolation functions and sensors locations, as well as of measurement errors, on the state estimation process. The method is demonstrated by means of two numerical examples.
The derivation of the equations of motion for the Spacecraft Control Laboratory Experiment (SCOLE) is reported and the equations of motion of a similar structure orbiting the earth are also derived. The structure is assumed to undergo large rigid-body maneuvers and small elastic deformations. A perturbation approach is proposed whereby the quantities defining the rigid-body maneuver are assumed to be relatively large, with the elastic deformations and deviations from the rigid-body maneuver being relatively small. The perturbation equations have the form of linear equations with time-dependent coefficients. An active control technique can then be formulated to permit maneuvering of the spacecraft and simultaneously suppressing the elastic vibration.
The objectives of this study are listed as follows: (1) to develop Lagrange's equations of motion for the shuttle antenna configuration in orbit; (2) to modify equations using the Lagrange multiplier method to develop equations of motion for the laboratory experiment; and (3) to discuss methods for simulation and control. The equations are presented in graph form.
An inherent incongruity in the modeling of controls for structures is discussed. Structures are basically distributed-parameter systems, described by partial differential equations, and control theory is concerned almost exclusively with discrete (in space) systems, described by ordinary differential equations. The standard approach to solving this dilemma is to discretize the system in space, which precludes the use of distributed controls. A different approach, known as the independent modal-space control method, is designed to eliminate the incongruity by bringing about a closer correspondence between modeling and control theory. The independent modal-space control method can treat distributed structures as well as discretized models and it permits design of both distributed and discrete-point controls.
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 independent modal space control (IMSC) method is described and compared with the coupled controls method. Although the coupled controls require fewer actuators, the IMSC provides: larger choice of control techniques, including nonlinear control; lower computational effort; lower computer storage requirement; lower control energy; provable robustness; and actuator location flexibility.
Results are reported from an experiment designed to control the vibratory motion of a beam at NASA, Langley Research Center. The experimental setup consists of a free-free uniform beam acted upon by four electromagnetic force actuators, with the motion being measured by nine displacement sensors. The entire assembly is linked to a CDC Cyber 175 computer which permits on-line real-time computation of the control forces. The control scheme is based on Independent Modal-Space Control (IMSC), whereby the modal forces are computed using a nonlinear, on-off control law. The actuator forces are then synthesized using a linear transformation, resulting in quantized forces. The sensors data is processed by modal filters. It is observed that the controls designed on the basis of the IMSC method are very effective in suppressing the vibratory motion of the beam, even though there is about a 50 percent differene between the actual and the analytically computed frequencies.
A scheme for the control and maneuvering of a large flexible spacecraft by means of two flexible AMCD's using noncontacting magnetic suspension is presented. The system consists of a flexible vehicle, two flexible rings and a magnetic suspension and driving assembly. The necessary skewing of the rings for maneuvering of the vehicle is accomplished by moving the pairs of magnets along tracks distributed around the circumference of the vehicle. The equations of motion for each subsystem are derived by the Lagrangian approach. Attitude motions are described in terms of quasi-coordinates. For small vehicle angular rates and rings attitude motions, an ordering scheme can be used to separate the equations of motion according to the magnitude of the terms. The ordered equations of motion lead to a linear time-variant optimal control problem for the maneuvering of the spacecraft.
The derivation of the force-free equations of motion of a spinning flexible ring via Lagrange's equations is presented. Closed-form expressions for the natural frequencies are obtained for the most general motion of the ring. Numerical results for rings of different radii and spin rates are presented.
Large flexible spacecraft represent highly complex distributed-parameter systems. In theory, a distributed-parameter system possesses an infinite number of degrees of freedom. For practical reasons, however, it must be discretized, which implies truncation. From a structural dynamics point of view, a mathematical model can be made more accurate by retaining an increasing number of degrees of freedom. From a control theory point of view, however, an increasing number of degrees of freedom places severe demands on the reliability of the various control algorithms. There is a difference of at least one order of magnitude between the structural modeling requirements and control theory capability. One of the challenges of large space structures control technology is to bridge this difference.
This paper advances a unified theory for the modal-space control method developed by these authors for the control of large flexible structures and places the theory on a more rigorous foundation. The unified approach permits the simultaneous treatment of a broad spectrum of interrelated problems. In particular, the theory is valid for both nongyroscopic and gyroscopic systems and for systems with ignorable coordinates resulting from rigid body motions. In addition, it permits a convenient way of studying the question of control spillover. The modal-space control method is different from any other modal control method in that it provides both internal (plant) and external (controller) decoupling.
This paper outlines a computer program especially tailored to the task of deriving explicit equations of motion for structures with point-connected substructures. The special purpose program is written in FORTRAN and is designed for performing the specific algebraic operations encountered in the derivation of explicit equations of motion. The derivation is by the Lagrangian approach. Using an orderly kinematical procedure and a discretization and/or truncation scheme, it is possible to write the kinetic and potential energy of each substructure in a compact vector-matrix form. Then, if each element of the matrices and vectors encountered in the kinetic and potential energy is a known algebraic expression, the computer program performs the necessary operations to evaluate the kinetic and potential energy of the system explicitly. Lagrange's equations for small motions about equilibrium can be deduced directly from the explicit form of the system kinetic and potential energy.
The dynamic synthesis of gyroscopic structures consisting of point-connected substructures is investigated. The objective is to develop a mathematical model capable of an adequate simulation of the modal characteristics of a helicopter using a minimum number of degrees of freedom. The basic approach is to regard the helicopter structure as an assemblage of flexible substructures. The variational equations for the perturbed motion about certain equilibrium solutions are derived. The discretized variational equations can be conveniently exhibited in matrix form, and a great deal of information about the system modal characteristics can be extracted from the coefficient matrices. The derivation of the variational equations requires a monumental amount of algebraic operations. To automate this task a symbolic manipulation program on a digital computer is developed.