Duality, observability, and controllability for linear time-varying descriptor systems

A characterization of observability for linear time-varying descriptor systemsE(t)x(t)+F(t)x(t)=B(t)u(t), y(t)=C(t)x(t) was recently developed. NeitherE norC were required to have constant rank. This paper defines a dual system, and a type of controllability so that observability of the original system is equivalent to controllability of the dual system. Criteria for observability and controllability are given in terms of arrays of derivatives of the original coefficients. In addition, the duality results of this paper lead to an improvement on a previous fundamental structure result for solvable systems of the formE(t)x(t)+F(t)x(t)=f(tt).

On the stability radius of a generalized state-space system

The concept of “distance to instability” of a system matrix is generalized to system pencils which arise in descriptor (semistate) systems. Difficulties arise in the case of singular systems, because the pencil can be made unstable by an infinitesimal perturbation. It is necessary to measure the distance subject to restricted, or structured, perturbations. In this paper a suitable measure for the stability radius of a generalized state-space system is defined, and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one. For systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles.

On the Convergence of Two-Stage Iterative Processes for Solving Linear Equations

This paper considers two-stage iterative processes for solving the linear system $Af = b$. The outer iteration is defined by $Mf^{k + 1} = Nf^k + b$, where $M$ is a nonsingular matrix such that $M - N = A$. At each stage $f^{k + 1} $ is computed approximately using an inner iteration process to solve $Mv = Nf^k + b$ for $v$. At the $k$th outer iteration, $p_k $ inner iterations are performed. It is shown that this procedure converges if $p_k \geqq P$ for some $P$ provided that the inner iteration is convergent and that the outer process would converge if $f^{k + 1} $ were determined exactly at every step. Convergence is also proved under more specialized conditions, and for the procedure where $p_k = p$ for all $k$, an estimate for $p$ is obtained which optimizes the convergence rate. Examples are given for systems arising from the numerical solution of elliptic partial differential equations and numerical results are presented.

Accommodative instability: relationship to progression of early onset myopia

Background: In a previous study, we demonstrated that children with early onset myopia had greater instability of accommodation than a group of emmetropic children. Since that study was correlational, we were unable to determine the causal relationship between this and myopic progression. To address this, we examined the children two years later. We predicted that if accommodative instability was causing the myopic progression, instability at Visit 1 should predict the refractive error at Visit 2. Additionally, instability at Visit 1 should predict myopic progression. Methods: Thirteen myopic and 16 emmetropic children were included in the analysis. Dynamic measures of accommodation were made using eccentric photorefraction (PowerRefractor) while children viewed targets set at three distances (accommodative demands), namely, 0.25 metres (4.00 D demand), 0.5 metres (2.00 D demand) and 4.00 metres (0.25 D demand). Results: Both refractive error and accommodative instability at Visit 1 were highly correlated with the same measures at Visit 2. Children with myopia showed greater instability of accommodation (0.38 D) than children with emmetropia (0.26 D) at the 4.00 D target on Visit 1 and this instability of accommodation weakly predicted myopic progression. Conclusions: The results presented in the present study suggest that instability of accommodation accompanies myopic progression, although a casual relationship cannot be established.