Optimal Hankel norm model reduction for discrete-time descriptor systems

• Published in: Journal of the Franklin Institute Vol. 356; no. 7; pp. 4124 - 4143
• Main Authors: Cao, X., Saltik, M.B., Weiland, S.
• Format: Journal Article
• Language: English
• Published: Elmsford Elsevier Ltd 01.05.2019; Elsevier Science Ltd
• Subjects: Algorithms; Approximation; Computer simulation; Discrete time systems; Dynamical systems; Estimating techniques; Mathematical analysis; Mathematical models; Model reduction; Numerical analysis; Numerical methods; Time invariant systems
• ISSN: 0016-0032; 1879-2693; 0016-0032
• DOI: 10.1016/j.jfranklin.2018.11.047

•Novel methods for optimal Hankel norm approximation of discrete-time descriptor systems are discussed.•Three Hankel operators are proposed. The associated Hankel norms are analyzed and the approximation errors are presented.•Index-1 systems are equivalent to state space systems without any algebraic coupling. The equivalent system is provided.•For high index case, structure-preserving low rank approximation of the finite Hankel matrix is the key mechanism. Optimal Hankel norm model reduction for dynamical systems is of great significance in model-based simulation and design. For the class of linear time-invariant systems, it is among the few optimal reduction methods for which a prior error bound between the original system and its approximation is known. However, for descriptor systems, this optimal approximation technique no longer applies. In this paper, we propose several definitions of the Hankel operator for dynamical discrete-time descriptor systems. We investigate the implications of these definitions for the problem of optimal model approximation of descriptor systems in the sense of the Hankel norm. Novel reduction algorithms are derived for this class of systems with and without preservation of the DAE-index. The performance of the proposed methods is illustrated by numerical examples.