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A matrix pencil approach to the Morgan's problem

Dimitris Vafiadis

TL;DR

This work advances Morgan's problem for nonsquare state-space systems by decomposing the original plant into a finite set of square subsystems generated via singular state feedback (SSF). Each subsystem is classified by controllability indices (CI) and tested for decouplability using established square-system criteria, with a constructive procedure to recover decoupling gains for the original system. A complete algorithm is presented that searches over admissible CI configurations, builds intermediate matrices and coordinate transforms, and yields decoupling pairs (or proves insolvability). The analysis also characterizes fixed poles, including input decoupling zeros, in terms of the chosen CI, and explains when these poles can be assigned. The approach provides a rigorous, finite, and systematic framework for diagonalizing the transfer function of nonsquare, right-invertible systems under SSF and regular input transformations.

Abstract

The problem of decoupling a nonsquare state space system by state feedback with singular input transformation is considered. The problem is solved by conducting a finite search for decouplable square systems, appropriately derived from the original. Decoupling feedback on any of these systems defines the decoupling feedback for the original. The issue of fixed poles is also considered and the possibility of selecting the uncontrollable poles is investigated.

A matrix pencil approach to the Morgan's problem

TL;DR

This work advances Morgan's problem for nonsquare state-space systems by decomposing the original plant into a finite set of square subsystems generated via singular state feedback (SSF). Each subsystem is classified by controllability indices (CI) and tested for decouplability using established square-system criteria, with a constructive procedure to recover decoupling gains for the original system. A complete algorithm is presented that searches over admissible CI configurations, builds intermediate matrices and coordinate transforms, and yields decoupling pairs (or proves insolvability). The analysis also characterizes fixed poles, including input decoupling zeros, in terms of the chosen CI, and explains when these poles can be assigned. The approach provides a rigorous, finite, and systematic framework for diagonalizing the transfer function of nonsquare, right-invertible systems under SSF and regular input transformations.

Abstract

The problem of decoupling a nonsquare state space system by state feedback with singular input transformation is considered. The problem is solved by conducting a finite search for decouplable square systems, appropriately derived from the original. Decoupling feedback on any of these systems defines the decoupling feedback for the original. The issue of fixed poles is also considered and the possibility of selecting the uncontrollable poles is investigated.
Paper Structure (6 sections, 4 theorems, 84 equations)

This paper contains 6 sections, 4 theorems, 84 equations.

Table of Contents

  1. Introduction
  2. Problem statement and background results
  3. Singular state feedback and input -- state pencil
  4. The solution of the problem
  5. Fixed poles
  6. Conclusions

Key Result

Theorem 2.1

The system if system (mp1) is square then it is decouplable by state feedback and regular input transformation if and only if the matrix $N_{\alpha} = [CS(s)\hbox{diag}\{s^{\sigma - \sigma_i}\}]_{hr}$, $\sigma = max\{\sigma_i\}, i =1, \cdots, \ell$ has full row rank. $\Box$

Theorems & Definitions (7)

  • Theorem 2.1
  • Theorem 3.1
  • Remark 3.1
  • Theorem 3.2
  • Theorem 4.1
  • Example 4.1
  • Example 5.1