This paper is a brief historical review of linear singular systems, followed by a survey of results on their solution and properties. The frequency domain and time domain approaches are discussed together to sketch an overall picture of the current status of the theory.

A survey of linear singular systems

Systems of the form E\dot{x}=Ax + Bu, y=Cx , with E singular, are studied. Of particular interest are the impulsive modes that may appear in the free-response of such systems when arbitrary initial conditions are permitted, modes that are associated with natural system frequencies at infinity. A generalized definition of system order that incorporates these impulsive degrees of freedom is proposed, and concepts of controllability and observability are defined for the impulsive modes. Allowable equivalence transformations of such singular systems are specified. The present framework is shown to overcome several difficulties inherent in other treatments of singular systems, and to extend, in a natural and satisfying way, many results previously known only for regular state-space systems.

A generalized state-space for singular systems

Lasaga's (1982) Master Equation for crystal growth is solved for multicomponent systems in situations which allow for coupled diffusion of melt species. The structure of the solution is explored in some detail for the case of a constant diffusion coefficient matrix. Incorporating these results, the growth of plagioclase is modeled in undercooled tholeiitic melts by approximating interface growth rates with (1) a reduced growth rate function and with (2) calculated solid-liquid solution properties obtained from the silicate liquid solution model of Ghiorso et al. (1983; appendix of Ghiorso 1985). For this purpose algorithms are provided for estimating the liquidus temperature or the chemical affinity of a multicomponent solid solution precipitating from a complex melt of specified bulk composition. Compositional trends in initial solids produced by successive degrees of undercooling are opposite to those predicted in the binary system NaAlSi3O8-CaAl2Si2O8. Calculations suggest that the solid phase and interface melt compositions rapidly approach a “steady state” for a given degree of undercooling. Consequently, the overall isothermal growth rate of plagioclase forming from tholeiitic melts appears to be entirely diffusion controlled. In magmatic systems the multicomponent growth equations allow for the formation of oscillatory zoned crystals as a consequence of the “couplingr” between interface reaction kinetics and melt diffusion. The magnitude of this effect is largely dependent upon the asymmetry of the diffusion coefficient matrix. Methods are described to facilitate the calibration of diffusion matrices from experimental data on multicomponent penetration curves.Experimental results (Lesher and Walker 1986) on steady state Soret concentration profiles resulting from thermal diffusion in MORB and andesitic liquids are analyzed using the theory of multicomponent linear irreversible thermodynamics. Under conditions where the entropy production is minimized, a linear relationship is derived between liquid chemical potentials and temperature. This relationship is utilized to evaluate the validity of the solution model of Ghiorso et al. (1983) in melts up to 300° C above their liquidus. The results indicate that configurational entropies are accurately modeled for MORB and andesite bulk compositions. The modeling fails in two four-component systems tested. Equations are derived which allow the calibration of multicomponent regular solution parameters from steady state Soret arrays. An algorithm is demonstrated which permits the calculation of steady state Soret concentration profiles, given an overall bulk melt composition and temperature gradient. This algorithm uses the liquid solution properties of Ghiorso et al. (1983) and constants obtained from the experimental measurements of Lesher and Walker (1986).

Chemical mass transfer in magmatic processes: III. Crystal growth, chemical diffusion and thermal diffusion in multicomponent silicate melts

The concepts of controllability and observability for systems of the form E\dot{x} = Ax + Bu, y = Cx, E singular are considered. A theory is presented which unifies the three main approaches to this topic already existing in the literature. The development includes a generalization of the duality theorem from state-space theory.

Controllability, observability, and duality in singular systems

Singular perturbations and time-scale methods in control theory: Survey 1976-1983

In this paper, we investigate the properties of the continuous descriptor system E\dot{x}(t)=Ax(t)+Bu(t), 0\leqt\leqb where E, A, and B are complex and possibly singular matrices and u(t) is a complex function differentiable sufficiently many times. The traditional approach to such systems is to separate the state equations from the algebraic equations. However, such algorithms usually destroy the natural, physically-based sparsity and structure of the original system. Therefore, we consider descriptor systems in their original form. Such systems possess numerous properties not shared by the well-known state variable systems. First, we relate classical theories of matrix pencils to the solvability of descriptor systems. Then we extend the concepts of reachability, controllability, and observability of state variable systems to descriptor systems, and describe the set of reachable states for descriptor systems.

Solvability, controllability, and observability of continuous descriptor systems

The numerical solution of physically realistic nonlinear partial differential equations (PDEs) is a complicated and highly problem-dependent process which usually requires the scientist to undertake the difficult and time-consuming task of developing his own computer program to solve his problem. This paper presents a software interface which can eliminate much of the expensive and time-consuming effort involved in the solution of nonlinear PDEs. The software interface provides centered differencing in the spatial variable for time-dependent nonlinear PDEs, giving a semidiscrete system of nonlinear ordinary differential equations (ODEs), which are then solved using one of the recently developed robust ODE integrators. Besides being portable, efficient, and easy to use, the software interface along with an ODE integrator will discretize the problem, select the time step and order, solve the nonlinear equations (checking for convergence, etc.), and maintain a user-specified time integration accuracy, all automatically and reliably. Physically realistic examples are given to illustrate the use and capability of the software.

Software for Nonlinear Partial Differential Equations

In this paper we analyze numerical methods for the solution of the large scale dynamical system E\dot{y}(t)=Ay(t)+g(t),Y(t_{0})=y_{0} , where E and A are matrices, possibly singular. Systems of this type have been referred to as implicit systems and more recently as descriptor systems since they arise from formulating system equations in physical variables. Special cases of such systems are algebraic-differential systems. We discuss the numerical advantages of this formulation and identify a class of numerical integration algorithms which have accuracy and stability properties appropriate to descriptor systems and which preserve structure, detect nonsolvable systems, resolve initial value consistency problems, and are applicable to "stiff" descriptor systems. We also present an algorithm for the control of the local truncation error on only the state variables.

Analysis of descriptor systems using numerical algorithms

In this paper we describe a software interface to solve time-dependent coupled systems of nonlinear partial differential equations (PDEs), which are defined over a two-dimensional rectangular rebqon. The software interface employs the method of lines technique whereby centered differencing with respect to the two spatial variables results in a system of time-dependent ordinary differential equations (ODEs), which can then be solved using one of the readily available modern ODE integrators. We also describe a routine to be used in conjunction with the interface for the efficient generation of the Jacobian matrix. Representative examples are given to demonstrate the use, validity, and capabihties of the accompanying algorithm.

Richard F. Sincovec

Papers

Solvability, controllability, and observability of continuous descriptor systems

Software for Nonlinear Partial Differential Equations

Analysis of descriptor systems using numerical algorithms

General Software for Two-Dimensional Nonlinear Partial Differential Equations

Proceedings of the Sixth Siam Conference on Parallel Processing for Scientific Computing