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Methods for the presentation and display of multivariate data are discussed with emphasis placed on the multivariate analysis of variance problems and the Hotelling T(2) solution in the two-sample case. The methods utilize the concepts of stepwise discrimination analysis and the computation of partial correlation coefficients.
The multivariable instrumental variable/approximate maximum likelihood (IV/AML) method or recursive time-series analysis is used to identify the multivariable (four inputs-three outputs) dynamics of the Pratt and Whitney F100 engine. A detailed nonlinear engine simulation is used to determine linear engine model structures and parameters at an operating point using open loop data. Also, the IV/AML method is used in a direct identification mode to identify models from actual closed loop engine test data. Models identified from simulated and test data are compared to determine a final model structure and parameterization that can predict engine response for a wide class of inputs. The ability of the IV/AML algorithm to identify useful dynamic models from engine test data is assessed.
Current efforts to develop methods and computer algorithms to effectively represent multivariate data commonly encountered in remote sensing applications are described. While this may involve scatter diagrams, multivariate representations of nonparametric probability density estimates are emphasized. The density function provides a useful graphical tool for looking at data and a useful theoretical tool for classification. This approach is called a thunderstorm data analysis.
The multivariable instrumental variable/approximate maximum likelihood (IV/AML) method of recursive time-series analysis is used to identify the multivariable (four inputs-three outputs) dynamics of the Pratt and Whitney F100 engine. A detailed nonlinear engine simulation is used to determine linear engine model structures and parameters at an operating point using open loop data. Also, the IV/AML method is used in a direct identification made to identify models from actual closed loop engine test data. Models identified from simulated and test data are compared to determine a final model structure and parameterization that can predict engine response for a wide class of inputs. The ability of the IV/AML algorithm to identify useful dynamic models from engine test data is assessed. Previously announced in STAR as N82-20339
The design of a multivariable feedback control system for the Pratt and Whitney F-100 turbofan jet engine is a challenging task for control engineers. This paper employs a linearized model of the F-100 engine to demonstrate the use of the newly developed Linear Quadratic Gaussian/Loop Transfer Recovery (LQG/LTR) design methodology, which adopts an integrated frequency-domain and time-domain approach to multivariable feedback control synthesis so as to meet stability-robustness, command-following, and disturbance-rejection specifications.
This paper considers the design of fixed order dynamic compensators for multivariable time invariant linear systems, minimizing a linear quadratic performance cost functional. Attention is given to robustness issues in terms of multivariable frequency domain specifications. An output feedback formulation is adopted by suitably augmenting the system description to include the compensator states. Either a controller or observer canonical form is imposed on the compensator description to reduce the number of free parameters to its minimal number. The internal structure of the compensator is prespecified by assigning a set of ascending feedback invariant indices, thus forming a Brunovsky structure for the nominal compensator.
An overview of several multivariate image processing techniques is presented, with emphasis on techniques based upon the principal component transformation (PCT). Multiimages in various formats have a multivariate pixel value, associated with each pixel location, which has been scaled and quantized into a gray level vector, and the bivariate of the extent to which two images are correlated. The PCT of a multiimage decorrelates the multiimage to reduce its dimensionality and reveal its intercomponent dependencies if some off-diagonal elements are not small, and for the purposes of display the principal component images must be postprocessed into multiimage format. The principal component analysis of a multiimage is a statistical analysis based upon the PCT whose primary application is to determine the intrinsic component dimensionality of the multiimage. Computational considerations are also discussed.
A three-dimensional, multivariate, statistical analysis method, optimal interpolation (OI) is described for modeling meteorological data from widely dispersed sites. The model was developed to analyze FGGE data at the NASA-Goddard Laboratory of Atmospherics. The model features a multivariate surface analysis over the oceans, including maintenance of the Ekman balance and a geographically dependent correlation function. Preliminary comparisons are made between the OI model and similar schemes employed at the European Center for Medium Range Weather Forecasts and the National Meteorological Center. The OI scheme is used to provide input to a GCM, and model error correlations are calculated for forecasts of 500 mb vertical water mixing ratios and the wind profiles. Comparisons are made between the predictions and measured data. The model is shown to be as accurate as a successive corrections model out to 4.5 days.
The paper presents a simple method for the design of linear multivariable controllers for multi-link robot manipulators. The control scheme consists of multivariable feedforward and feedback controllers. The feedforward controller is the minimal inverse of the linearized model of robot dynamics and contains only proportional-double-derivative (PD2) terms. This controller ensures that the manipulator joint angles track any reference trajectories. The feedback controller is of proportional-integral-derivative (PID) type and achieves pole placement. This controller reduces any initial tracking error to zero as desired and also ensures that robust steady-state tracking of step-plus-exponential trajectories is achieved by the joint angles. The two controllers are independent of each other and are designed separately based on the linearized robot model and then integrated in the overall control scheme. The proposed scheme is simple and can be implemented for real-time control of robot manipulators.
The paper presents a simple scheme for the linear multivariable control of multi-link robot manipulators. The control scheme consists of multivariable feedforward and feedback controllers. The feedforward controller is the minimal inverse of the linearized model of robot dynamics and contains only proportional-double-derivative terms. This controller ensures that the manipulator joint angles track any reference trajectories. The feedback controller is of proportional-derivative type and achieves stability and pole placement. This controller reduces any initial tracking error to zero as desired. The two controllers are independent of each other and are designed separately based on the linearized robot model and then integrated in an overall control scheme.
The paper presents simple schemes for multivariable control of multiple-joint robot manipulators in joint and Cartesian coordinates. The joint control scheme consists of two independent multivariable feedforward and feedback controllers. The feedforward controller is the minimal inverse of the linearized model of robot dynamics and contains only proportional-double-derivative (PD2) terms - implying feedforward from the desired position, velocity and acceleration. This controller ensures that the manipulator joint angles track any reference trajectories. The feedback controller is of proportional-integral-derivative (PID) type and is designed to achieve pole placement. This controller reduces any initial tracking error to zero as desired and also ensures that robust steady-state tracking of step-plus-exponential trajectories is achieved by the joint angles. Simple and explicit expressions of computation of the feedforward and feedback gains are obtained based on the linearized model of robot dynamics. This leads to computationally efficient schemes for either on-line gain computation or off-line gain scheduling to account for variations in the linearized robot model due to changes in the operating point. The joint control scheme is extended to direct control of the end-effector motion in Cartesian space. Simulation results are given for illustration.
The design of feasible controllers for high dimension multivariable systems can be greatly aided by a method of model reduction. In order for the design based on the order reduction to include a guarantee of stability, it is sufficient to have a bound on the model error. Previous work has provided such a bound for continuous-time systems for algorithms based on balancing. In this note an L-infinity bound is derived for model error for a method of order reduction of discrete linear multivariable systems based on balancing.
A new methodology is developed for the synthesis of linear, time-invariant (LTI) controllers for multivariable LTI systems. The aim is to achieve stability and performance robustness of the feedback system in the presence of multiple unstructured uncertainty blocks; i.e., to satisfy a frequency-domain inequality in terms of the structured singular value. The design technique is referred to as the Causality Recovery Methodology (CRM). Starting with an initial (nominally) stabilizing compensator, the CRM produces a closed-loop system whose performance-robustness is at least as good as, and hopefully superior to, that of the original design. The robustness improvement is obtained by solving an infinite-dimensional, convex optimization program. A finite-dimensional implementation of the CRM was developed, and it was applied to a multivariate design example.
AUTOCON is an automated computer-aided design tool for the synthesis and optimization of linear multivariable control systems based upon user-defined control parameter optimization. Violations in stability and performance requirements are computed from constraints on Single Input/Single Output (SISO) open- and closed-loop transfer function frequency responses, and from constraints on the singular-value frequency responses of Multiple Input/Multiple Output (MIMO) transfer functions, for all critical plant variations. Optimum nonlinear programming algorithms are used in the search for local constrained solutions in which violations in stability and performance are caused either to vanish or be minimized for a proper selection of the control parameters. Classical control system stability and performance design can, in this way, be combined with modern multivariable robustness methods to offer general frequency response loop-shaping via a computer-aided design tool. Complete Nichols, Nyquist, Bode, singular-value Bode magnitude and transient response plots are produced, including user-defined boundary responses. AUTOCON is used to synthesize and optimize the lateral/directional flight control system for a typical high-performance aircraft.
Many approaches in the field of linear, multivariable time-invariant systems analysis and controller synthesis employ loop-sharing procedures wherein design parameters are chosen to shape frequency-response singular value plots of selected transfer matrices. A software package, FREQ, is documented for computing within on unified framework many of the most used multivariable transfer matrices for both continuous and discrete systems. The matrices are evaluated at user-selected frequency-response values, and singular values against frequency. Example computations are presented to demonstrate the use of the FREQ code.
A new methodology is developed for the synthesis of linear, time-invariant (LTI) controllers for multivariable LTI systems. The aim is to achieve stability and performance robustness of the feedback system in the presence of multiple unstructured uncertainty blocks; i.e., to satisfy a frequency-domain inequality in terms of the structured singular value. The design technique is referred to as the causality recovery methodology (CRM). Starting with an initial (nominally) stabilizing compensator, the CRM produces a closed-loop system whose performance-robustness is at least as good as, and hopefully superior to, that of the original design. The robustness improvement is obtained by solving an infinite-dimensional, convex optimization program. A finite-dimensional implementation of the CRM was developed, and it was applied to a multivariate design example.
The present treatment of the related problems of minimal inversion and perfect output control in linear multivariable systems uses a simple analytical expression for the inverse of a square multivariate system's transfer-function matrix to construct a minimal-order inverse of the system. Because the poles of the minimal-order inverse are the transmission zeros of the system, necessary and sufficient conditions for the inverse system's stability are simply stated in terms of the zero polynomial of the original system. A necessary and sufficient condition for the existence of the required controllers is that the plant zero polynomial be neither identical to zero nor unstable.
A six-input/six-output multivariable adaptive controller is used to control a large 3-dimensional flexible structure experiment. For this purpose, the JPL/AFAL antenna-like experiment structure is used, with instrumentation distributed on both the hub and the ribs. This represents a significant increase in complexity relative to earlier adaptive control experiments performed on the same structure, utilizing a two-input/two-output set-up with instrumentation on the hub alone. The increase in instrumentation significantly increases spatial controllability, and excitation and control of a much larger set of modes. This paper documents multivariable adaptive control experiments on this structural configuration, and discusses many of the associated design and implementation issues required for successful operation.