[Paper Review] Descriptor System Tools (DSTOOLS) User's Guide

Andreas Varga|arXiv (Cornell University)|Jul 22, 2017

Fault Detection and Control Systems56 references3 citations

TL;DR

This paper presents DSTOOLS, a MATLAB-based software toolbox for analyzing and manipulating rational transfer function matrices using descriptor system representations. It enables advanced control system operations such as model order reduction, coprime factorizations, Hankel norm computations, and gap metric analysis, with enhanced numerical reliability and stability boundary detection for poles and zeros.

ABSTRACT

The Descriptor System Tools (DSTOOLS) is a collection of MATLAB functions for the operation on and manipulation of rational transfer function matrices via their descriptor system realizations. The DSTOOLS collection relies on the Control System Toolbox and several mex-functions based on the Systems and Control Library SLICOT. Many of the implemented functions are based on the computational procedures described in Chapter 10 of the book: "A. Varga, Solving Fault Diagnosis Problems - Linear Synthesis Techniques, Springer, 2017". This document is the User's Guide for the version V0.71 of DSTOOLS. First, we present the mathematical background on rational matrices and descriptor systems. Then, we give in-depth information on the command syntax of the main computational functions. Several examples illustrate the use of the main functions of DSTOOLS.

Motivation & Objective

• To provide a comprehensive, numerically reliable toolbox for manipulating rational transfer function matrices via descriptor system realizations.
• To support advanced control system operations such as model order reduction, coprime factorizations, and system balancing.
• To improve numerical stability and accuracy in detecting poles and zeros on the boundary of the stability domain.
• To enable robust computation of system norms, gap metrics, and solutions to optimal control problems like Nehari and model matching.
• To ensure backward compatibility while introducing enhanced functionality for generalized linear time-invariant (LTI) systems.

Proposed method

• Utilizes descriptor system representations to model rational transfer function matrices, enabling analysis of improper and singular systems.
• Employs SLICOT-based mex-functions for high-performance numerical computation of system properties and transformations.
• Implements advanced algorithms for computing generalized poles, zeros, and nullspace bases using structured matrix pencils.
• Introduces offset parameters in key functions (e.g., gpole, gzero) to improve robustness in detecting boundary poles and zeros.
• Applies balancing-related and coprime factorization techniques for model order reduction and system approximation.
• Supports inner-outer and spectral factorizations, along with solutions to optimal Nehari and least-distance problems via Riccati equation solvers.

Experimental results

Research questions

• RQ1 How can descriptor system representations be effectively used to compute and analyze rational transfer function matrices in a numerically stable way?
• RQ2 What computational methods ensure reliable detection of poles and zeros on the stability domain boundary, especially in the presence of numerical uncertainties?
• RQ3 How can generalized coprime factorizations and spectral factorizations be computed efficiently for robust control design?
• RQ4 What improvements can be made to model order reduction techniques for descriptor systems to preserve system properties like stability and controllability?
• RQ5 How can the ν-gap metric be computed reliably for arbitrary (including improper) LTI systems using descriptor-based formulations?

Key findings

• The DSTOOLS V0.71 release introduces the new function `gnugap` for computing the ν-gap metric, applicable to arbitrary LTI systems including improper ones.
• Functions like `gpole` and `gzero` now provide extensive structural information on the underlying matrix pencils and enhanced detection of boundary poles and zeros.
• The `gnrank` function replaces the deprecated `nrank`, with backward compatibility preserved for legacy code.
• Enhancements to `grnull`, `glnull`, `grange`, and `gcrange` include optional inner/coinner nullspace basis computation and improved rank and stability information via the `INFO` structure.
• The `gsklf` function now reports the number of finite and infinite unstable zeros in continuous-time systems, with corrected detection of impulse controllability.
• Bug fixes in `grasol`, `glasol`, and `glinfldp` improve handling of empty systems and convergence in γ-iteration, enhancing robustness in optimal control computations.