In this paper we give a survey on balanced truncation model order reduction for linear time-invariant continuous-time systems in descriptor form. We first give a brief overview of the basic concepts from linear system theory and then present balanced truncation model reduction methods for descriptor systems and discuss their algorithmic aspects. The efficiency of these methods is demonstrated by numerical experiments.

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Balanced Truncation Model Reduction for Large-Scale Systems in Descriptor Form

Differential Algebraic Systems In order to treat coupled systems of partial differential equations (PDEs) and differential-algebraic equations (DAEs) in a systematic way we shall study abstract differential algebraic systems (ADASs) of the following form A d dt D(u(t), t) + B(u(t), t) = 0 for t ∈ [t0, T ]. (4.1) This equation is to be understood as an operator equation with operators A, D(·, t) and B(·, t) acting in real Hilbert spaces. More precisely, let X, Y , Z ⊆ Z̃ be Hilbert spaces and A : Z̃ → Y, D(·, t) : X → Z, B(·, t) : X → Y. (4.2) In [Mär02a, LMT01], the case Z = Z̃ has been considered. This is enough for the classical formulation of the coupled system (see Section 3.5). However, for the generalized formulation, we need the more general case Z ⊆ Z̃ (see Section 3.6). Such systems with A and D being invertible have already been studied in [GGZ74]. However, the classical formulation as well as the generalized formulation of the coupled system lead to abstract differential algebraic systems with operators A or D(·, t) that are not invertible on the whole time interval. Such systems are also called singular (see e.g. [Cam80, Cam82] for the finite dimensional case) or degenerate differential equations (see e.g. [Kur93, FY99]). 66 Abstract Differential Algebraic Systems In [FY99], systems of the form (4.1) with linear operators have been studied. More precisely, initial value systems of the form A d dt Du + Bu = Ag(t) for t ∈ (0, T ], (4.3) Du(0) = v0 ∈ imD, (4.4) with X = Y = Z, D = M and A = I are treated by a semigroup approach. I denotes the identity operator. The system (4.3)-(4.4) is reduced to multivalued differential equations of the form dv dt ∈ Av + f(t) for t ∈ (0, T ], (4.5) v(0) = v0, (4.6) where A := −BD. Here, the operator D is defined as (multi-valued) function satisfying Dv = {u ∈ DD : Du = v} for all v ∈ imD. with DD being the definition domain of D. The existence and uniqueness of classical solutions of (4.3)-(4.4) satisfying Du ∈ C([0, T ]; X) and Bu ∈ C([0, T ]; X) is shown provided that g ∈ C([0, T ]; X), Bu(0) ∈ imA, Re(−Bu|Du)X ≤ β‖Du‖ 2 X , for all u ∈ DB ⊆ DD (4.7) as well as the operator λ0AD + B : DB → X is bijective for some λ0 > β. (4.8) A similar result is obtained for systems of the form (4.3)-(4.4) with A = D or D = I. This approach via multi-valued differential equations concentrates on the dynamic part of the system. It is limited to systems with special constraints. So, for instance, condition (4.7) implies Re(Bw,Dv) = 0 for all w ∈ kerD ∩ DB, v ∈ DD. Together with condition (4.8), we obtain, for the finite dimensional case, a DAE of index 1 with the constraint (BQ)Bu = (BQ)g(t)

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Coupled Systems of Differential Algebraic and Partial Differential Equations in Circuit and Device Simulation

Publisher Summary This chapter discusses the modeling aspect of differential-algebraic equations (DAEs). In computational engineering, the network modeling approach forms the basis for computer-aided analysis of time-dependent processes in multibody dynamics, process simulation, or circuit design. Its principle is to connect compact elements via ideal nodes, and to apply some kind of conservation rules for setting up equations. The mathematical model, a set of so-called network equations, is generated automatically by combining network topology with characteristic equations describing the physical behavior of network elements under some simplifying assumptions. Interconnects and semiconductor devices (i.e., transistors) are modeled by multi-terminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities. Interconnects and semiconductor devices (i.e., transistors) are modeled by multiterminal elements (multi-ports), for which the branch currents entering any terminal and the branch voltages across any pair of terminals are well-defined quantities.

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Modelling and discretization of circuit problems

Stochastic differential algebraic equations of index 1 and applications in circuit simulation

We generalize an alternating direction implicit method and the Smith method for 
large-scale projected generalized Lyapunov equations. Such equations arise in model reduction for 
descriptor systems. Low rank versions of these methods are also presented, that can be used to 
compute low rank approximations to the solution of projected generalized Lyapunov equations with 
low rank symmetric, positive semidefinite right-hand side. Numerical examples are presented.

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Low rank iterative methods for projected generalized Lyapunov equations

The development of integrated circuits requires powerful numerical simulation programs. Naturally, there is no method that treats all the different kinds of circuits successfully. The numerical simulation tools provide reliable results only if the circuit model meets the assumptions that guarantee a successful application of the integration software. Owing to the large dimension of many circuits (about 107 circuit elements) it is often difficult to find the circuit configurations that lead to numerical difficulties. In this paper, we analyse electric circuits with respect to their structural properties in order to give circuit designers some help for fixing modelling problems if the numerical simulation fails. We consider one of the most frequently used modelling techniques, the modified nodal analysis (MNA), and discuss the index of the differential algebraic equations (DAEs) obtained by this kind of modelling. Copyright © 2000 John Wiley & Sons, Ltd.

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Structural analysis of electric circuits and consequences for MNA

One of the difficulties of the numerical integration methods for differential–algebraic equations (DAEs) is the computation of consistent initial values before starting the integration, i.e. calculating values that satisfy the given algebraic constraints as well as the hidden constraints if higher index problems are considered.



This paper presents an approach to calculate consistent initial values for index-2 DAEs starting up from possibly inconsistent ones for systems arising from modified nodal analysis (MNA) in circuit simulation. This article is based on the results from Estevez Schwarz and Tischendorf, International Journal of Circuit Theory and Applications 2000; 28: 131–162. Several of the denotations and results that we use were introduced there in more detail. Copyright © 2001 John Wiley & Sons, Ltd.

A step‐by‐step approach to compute a consistent initialization for the MNA

Circuit simulation is a standard task for the computer-aided design of electronic circuits. The transient analysis is well understood and realized in powerful simulation packages for conventional circuits. However, further developments in production engineering lead to new classes of circuits, which cause difficulties for the numerical integration. The dimension of circuit models can be quite large (10 5 equations). The complexity of the models demands a higher level of ion. Parasitic effects become dominant. The signal to noise ratio becomes smaller. In this paper, we want to draw attention to three essential problems from a mathematical point of view, the DAE-index, consistent initial values, and asymptotic stability. These topics have been extensively analyzed only recently. We shall illustrate them by some simple examples.

D. Estévez Schwarz

Papers

Structural analysis of electric circuits and consequences for MNA

A step‐by‐step approach to compute a consistent initialization for the MNA

Mathematical Problems in Circuit Simulation