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Gramian-based model reduction for descriptor systems. (English) Zbl 1067.93011
Linear time-invariant continuous-time systems like, \[ \begin{aligned} Ex(t) &= Ax(t)+ Bu(t),\;x(0)= x(0),\\ y(t) &= Cx(t),\end{aligned}\tag{\(*\)} \] where \(E\) and \(A\) are matrices in \(\mathbb R^{(n,n)}\), \(B\) in \(\mathbb R^{(n,m)}\), \(C\) in \(\mathbb R^{(p,n)}\), and \(x(t)\) and \(u(t)\) being vectors in \(\mathbb R^n\), and \(\mathbb R^p\) respectively, arise frequently, e.g. from spatial discretization of partial differential equations. Here, the order \(n\) of the system is usually large giving rise to some difficulties for simulation or real-time controller design because of storage requirements and expensive computations. A known technique to overcome these drawbacks is the so-called “model order reduction” consisting in approximating \((*)\) by a reduced-order system so that properties of the original system like regularity and stability are preserved. This approach has been successfully applied when \(E= I\) (standard state space systems). The more general problem involving an arbitrary matrix \(E\) (descriptor systems) is considered in the present paper.
In this respect, controllability and observability Gramians as well as Hankel singular values for descriptor systems are first introduced, and then an extension of the balanced truncation methods is presented by computing the generalized Schur form of the pencil \(zE- A\) (\(z\) a complex number) and solving the generalized Sylvester and Lyapunov equations. Illustrative numerical examples are also given.

93B11 System structure simplification
93B60 Eigenvalue problems
34A09 Implicit ordinary differential equations, differential-algebraic equations
93B05 Controllability
93B07 Observability
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