Conic Sector Research Articles

Notes on the Mathematics of the PHIGS Viewing Pipeline

AbstractIt is well known that homogeneous linear matrix transformations of the projective space may be efficiently used for the representation and the execution of common geometrical transformations but the general class of such transformations include also matrices which may cause numerical problems by transforming certain finite geometrical objects to infinity Different methods have been developed for handling such cases and this paper presents a new one called‘UW clipping’which is based on some interesting properties of projective transformations. Furthermore the paper introduces die concept of ‘conic sectors’ as a generalisation to half‐planes and halfspaces respectively, with the invariance property that such sectors are mapped onto other such sectors by projective transformations, and thus enable the transformation of clipping halfplanes and halfspaces. Finally the possibility of transforming die rectangular clipping box into the object space is investigated.

Robust design of adaptive control systems using conic sector theory

This paper presents a design approach for discrete adaptive control systems which provides a quantitative measure of the effect of design alternatives such as (i) adaptive gain, (ii) model order, and (iii) sampling rate, on stability in the presence of unmodeled plant dynamics. The proposed method, based on the conic conditions developed by Ortega et al. (1985, IEEE Trans. Aut. Control, AC-30, 1179–1187), is illustrated using a benchmark example. The results demonstrate that the sector conditions permit design tradeoffs to be made such that stability is maintained despite the model-plant mismatch.

New conic sectors for sampled-data feedback systems

New conic sectors are presented which can be used to analyze sampled-data feedback systems. First, a cone is constructed which contains just the sampler. Next, by using conic sector results for interconnected systems, the cone is propagated to the entire sampled-data operator. The new cone result is less conservative (has a smaller radius) and removes the restriction of an open-loop stable sampled-data compensator.

Conic sectors for sampled-data feedback systems

A multivariable analog system can be controlled by a sampled-data compensator. A conic sector that can be used to analyze the closed-loop stability and robustness of this feedback system is presented in this letter.

Propagation of conic model uncertainty in hierarchical systems

Conic sectors are used for analysis of stability and modeling uncertainty propagation in interconnected systems with a hierarchical structure. For large systems that can be readily decomposed into sparsely interconnected groups subsystems, sub-subsystems, sub-sub-subsystems, etc., analysis is greatly facilitated: formulas for conic uncertainty propagation and stability analysis are applied successively to the various groups, beginning at the bottom of the hierarchy and moving up one level at a time. At each level of the hierarchy the results both test stability and yield tight cone bounds on the modeling uncertainty in terms of the cone parameters at the level immediately below.

Propagation of conic model uncertainty in hierarchical systems

Conic sectors are used for analysis of stability and modeling uncertainty propagation in interconnected systems with a hierarchical structure. For large systems that can be readily decomposed into sparsely interconnected groups subsystems, sub-subsystems, sub-sub-subsystems, etc., analysis is greatly facilitated: formulas for conic uncertainty propagation and stability analysis are applied successively to the various groups, beginning at the bottom of the hierarchy and moving up one level at a time. At each level of the hierarchy the results both test stability and yield tight cone bounds on the modeling uncertainty in terms of the cone parameters at the level immediately below.

Propagation of conic model uncertainty in hierarchical systems

Conic sectors are used for analysis of stability and modeling uncertainty propagation in interconnected systems with a hierarchical structure. For large systems that can be readily decomposed into sparsely interconnected groups subsystems, sub-subsystems, sub-sub-subsystems, etc., analysis is greatly facilitated: formulas for conic uncertainty propagation and stability analysis are applied successively to the various groups, beginning at the bottom of the hierarchy and moving up one level at a time. At each level of the hierarchy the results both test stability and yield tight cone bounds on the modeling uncertainty in terms of the cone parameters at the level immediately below.

A multiloop generalization of the circle criterion for stability margin analysis

In order to provide a theoretical tool well suited for use in characterizing the stability margins (e.g., gain and phase margins) of multiloop feedback systems, multiloop input-output stability results generalizing the circle stability criterion are considered. Generalized conic sectors with centers and radii determined by linear dynamical operators are employed to enable an engineer to specify the stability margins which he desires as a frequency-dependent convex of modeling errors-including nonlinearities, gain variations, and phase variations-which the system must be able to tolerate in each feedback loop without instability. The resulting stability criterion gives sufficient conditions for closed-loop stability in the presence of such frequency-dependent modeling errors, even when the modeling errors occur simultaneously in all loops; so, for example, stability is assured as loop gains and phases vary throughout a set of nonzero whose boundaries are frequency-dependent. The stability conditions yield an easily interpreted scalar measure of the amount by which a muitiloop system exceeds, or falls short of, its stability margin specifications.

Input-Output Stability and Response Analysis for Hierarchical Large-Scale Systems

Analysis of hierarchially structured, possibly nonlinear, large scale systems is found to be greatly simplified using a result that permits testing for stability and generation of conic sector bounds for the I/O relation of an interconnected system from conic sector bounds on the subsystems I/O relations. For large scale systems that can be hierarchially decomposed into groups of subsystems, sub-subsystems, sub-sub-subsystems, et cetera so that each group has relatively few interconnections with other groups, analysis is greatly facilitated: the result is applied successively to the various groups, beginning by determining stability and conic sector bounds for groups at the bottom of the hierarchy, then doing likewise for sub-subsystems, subsystems and so forth until stability and conic sector bounds have been determined for the complete interconnected system.