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Mathematical models of a 5, 6, 7, and 8 coil large gap magnetic suspension system (MSDS) are presented. Some of the topics covered include: force and torque equations, reduction of state-space form, natural modes, origins of modes, effect of rotation in azimuth (yaw), future work, and n-coil ring conclusions.
The difference equations of Kalman filtering and smoothing recursively factor and invert the covariance of the output of a linear state-space system driven by a white-noise process. Here it is shown that similar recursive techniques factor and invert the inertia matrix of a multibody robot system. The random field models are based on the assumption that all of the inertial (D'Alembert) forces in the system are represented by a spatially distributed white-noise model. They are easier to describe than the models based on classical mechanics, which typically require extensive derivation and manipulation of equations of motion for complex mechanical systems. With the spatially random models, more primitive locally specified computations result in a global collective system behavior equivalent to that obtained with deterministic models. The primary goal of applying random field estimation is to provide a concise analytical foundation for solving robot control and motion planning problems.
A sensitivity analysis of flutter response of a two-degree of freedom airfoil with plunging and pitching degrees of freedom in transonic flow was performed using a state-space representation of the unsteady aerodynamic behavior. The structural equations of motion of the airfoil with bending and torsional degrees of freedom are coupled to the unsteady airloads, and the aeroelastic system so modeled is solved as an eigenvalue problem to determine the stability. The results of eigenanalysis showed good agreement with flutter calculations performed using a time-integration of the aeroelastic equations. The sensitivities of the flutter speed with respect to the mass and stiffness parameters wee computed by both the analytical and finite difference methods, showing excellent agreement.
A method is proposed by which an aeroservoelastic problem is brought to a state-space form with a minimum number of augmented aerodynamic terms. The examples treated in this work relate to NASA's Drone for Aerodynamic and Structural Testing-Aerodynamic Research Wing 1 (DAST-ARW1) and to the YF-17 fighter model. It is shown that in all cases considered, the method yields a very good accuracy regarding the flutter parameters and the dynamic behavior of the systems, using only two augmented aerodynamic states. The method should prove useful in the design of lower order control laws based on optimal control theory.
The computer programs and derivations generated in support of the modeling and design optimization program are presented. Programs for the buck regulator, boost regulator, and buck-boost regulator are described. The computer program for the design optimization calculations is presented. Constraints for the boost and buck-boost converter were derived. Derivations of state-space equations and transfer functions are presented. Computer lists for the converters are presented, and the input parameters justified.
Initial experiments on state-space feedback control of a large flexible manipulator with a parallel linkage drive are described. A linear controller using joint angle and strain measurements has been designed to minimize a quadratic performance index with a prescribed stability margin. It is based on a simplified model that accounts for the constraints of the parallel linkage kinematically rather than through constraint forces. The results show substantial improvement over a simple proportional-derivative joint control.
This paper discusses the effects of variations in system parameters on the dynamic behavior of the Free-Piston Stirling Engine/Linear Alternator (FPSE/LA)-load system. The mathematical formulations incorporate both the mechanical and thermodynamic properties of the FPSE, as well as the electrical equations of the connected load. A state-space technique in the frequency domain is applied to the resulting system of equations to facilitate the evaluation of parametric impacts on the system dynamic stability. Also included is a discussion on the system transient stability as affected by sudden changes in some key operating conditions. Some representative results are correlated with experimental data to verify the model and analytic formulation accuracies. Guidelines are given for ranges of the system parameters which will ensure an overall stable operation.
This paper discusses the effects of a variations in system parameters on the dynamic behavior of a Free-Piston Stirling Engine/Linear Alternator (FPSE/LA)-load system. The mathematical formulations incorporates both the mechanical and thermodynamic properties of the FPSE, as well as the electrical equations of the connected load. State-space technique in the frequency domain is applied to the resulting system of equations to facilitate the evaluation of parametric impacts on the system dynamic stability. Also included is a discussion on the system transient stability as affected by sudden changes in some key operating conditions. Some representative results are correlated with experimental data to verify the model and analytic formulation accuracies. Guidelines are given for ranges of the system parameters which will ensure an overall stable operation.
The Tracking and Data Relay Satellite System (TDRSS) uses a groundbased attitude determination algorithm to open loop point the satellite's high data rate antennas. The spacecraft is able to measure its pitch and roll attitude, but its yaw attitude is periodically unobservable. The ground software uses a state-space estimator, an adaptation of a Luenberger observer, to predict the spacecraft yaw angle during these unobservable periods. It contains states associated with the roll/yaw dynamics and the on-board control law. The accuracy is limited by the modeling fidelity of the disturbance torques acting on the spacecraft. After initial operating problems were cleared up, the operation of the estimator has converged to predicted performance.
Several methods are presented for defining generalized state space models (GSSM), with emphasis on 'fast' techniques for transforming GSSMs to condensed state space models (SSM). The fast forms are configured to yield invariant transfer functions. Details of the decomposition process are summarized in terms of the determination of eigenvalues which are separated within staircase matrices. Applications of the techniques are illustrated through implementation of a unimodular transformation which does not affect the finite eigenvalues, by demonstrating deadbeat control of a GSSM, which the definition of a reduced observer of a GSSM, and by embedding a polynomial matrix into a unimodular matrix.
Various control analysis, design, and simulation techniques for aeroelastic applications require the equations of motion to be cast in a linear time-invariant state-space form. Unsteady aerodynamics forces have to be approximated as rational functions of the Laplace variable in order to put them in this framework. For the minimum-state method, the number of denominator roots in the rational approximation. Results are shown of applying various approximation enhancements (including optimization, frequency dependent weighting of the tabular data, and constraint selection) with the minimum-state formulation to the active flexible wing wind-tunnel model. The results demonstrate that good models can be developed which have an order of magnitude fewer augmenting aerodynamic equations more than traditional approaches. This reduction facilitates the design of lower order control systems, analysis of control system performance, and near real-time simulation of aeroservoelastic phenomena.
A series of techniques are presented for analyzing airplane control and output characteristics. State-space matrix equations describing the linear perturbation dynamics are transformed from physical coordinates into scaled, modal coordinates. Techniques for analyzing the impacts of system inputs on the fundamental modes of motion and the appearance of these modes in the system outputs are explained. Scaled modal matrices are used to compute steady-state control inputs which optimize the steady-state response of selected system outputs. Graphics which promote quick understanding of the analysis are presented to display the resulting vectors and matrices. The defined analysis techniques are applied to an example airplane model, illustrating the insight which can be acquired using the described modal techniques.
Both classical and state-space synthesis methods for active control of flexible space structures in the presence of persistent disturbances are presented. The methods exploit the so-called internal model principle for asymptotic disturbance rejection. A generic example of flexible space structures is used to illustrate the simplicity of the proposed design methodologies. The concept of a disturbance rejection filter dipole is introduced from a classical control viewpoint. It is shown that the proposed design methods will invariably make use of non-minimum-phase compensation for a class of noncolocated control problems. The need for tradeoffs between performance and parameter robustness is discussed.
An engineering approach for analyzing airplane control and output characteristics is presented. State-space matrix equations describing the linear perturbation dynamics are transformed from physical coordinates into scaled coordinates. The scaling is accomplished by applying various transformations to the system to employ prior engineering knowledge of the airplane physics. Two different analysis techniques are then explained. Modal analysis techniques calculate the influence of each system input on each fundamental mode of motion and the distribution of each mode among the system outputs. The optimal steady state response technique computes the blending of steady state control inputs that optimize the steady state response of selected system outputs. Analysis of an example airplane model is presented to demonstrate the described engineering approach.
One of the major problems associated with the control of flexible structures is the estimation of system states. Since the parameters of the structures are not constant under varying loads and conditions, conventional fixed parameter state estimators can not be used to effectively estimate the states of the system. One alternative is to use a state estimator which adapts to the condition of the system. One such estimator is the Kalman filter. This filter is a time varying recursive digital filter which is based upon a model of the system being measured. This filter adapts the model according to the output of the system. Previously, the Kalman filter has only been used in an off-line capacity due to the computational time required for implementation. With recent advances in computer technology, it is becoming a viable tool for use in the on-line environment. A distributed Kalman filter implementation is described for fast estimation of the state of a flexible arm. A key issue, is the sensor structure and initial work on a distributed sensor that could be used with the Kalman filter is presented.
A new p-type, finite-state, aerodynamic theory is used in the aeroelastic analyses of a fixed wing and a rotating wing in hover. The helicopter inflow theory is modified for application to a fixed-wing, or stopped-rotor configuration. The structural models are linear for both configurations, with elastic bending and torsion degrees of freedom. Results are compared with those using Theodorsen and Loewy theories, respectively, for the fixed and rotating wings. With the identification of both structural and nerodynamic modes, the involved frequency content of the flow is seen in a new, more detailed perspective.
The computation-time delay in the feedback controller of a real-time control system may cause failure to update the control input during one or more sampling periods. If this delay exceeds a certain limit called a hard deadline, either the necessary conditions for system stability are violated or the system leaves the allowed state-space. In such a case a dynamic failure is said to occur to the system. A method for calculating the hard deadlines in linear time-invariant control systems by considering system stability and the allowed state-space is presented. To derive necessary conditions for (asymptotic) system stability, the state difference equation is modified based on an assumed maximum delay and the probability distribution of delays whose magnitudes are less than, or equal to, the assumed maximum delay. Moreover, the allowed state-space - which is derived from given input and state constraints - is used to calculate the hard deadline as a function of time and the system state. A one-shot delay model in which a single event causes a dynamic failure is also considered. The knowledge of hard deadline is then applied to the design of error recovery in a triple modular redundant (TMR) controller computer.
Report presents theoretical study of some issues of controllability and observability of system represented by linear, time-invariant mathematical model of the form. x = Ax + Bu, y = Cx + Du, x(0) = xo where x is n-dimensional vector representing state of system; u is p-dimensional vector representing control input to system; y is q-dimensional vector representing output of system; n,p, and q are integers; x(0) is intial (zero-time) state vector; and set of matrices (A,B,C,D) said to constitute state-space representation of system.