### Abstract

This paper represents a first attempt to derive a closed-form (Hankel-norm) optimal solution for multivariable system reduction problems. The basic idea is to extend the scalar case approach in [5] to deal with the mulrivariable systems. The major contribution lies in the development of a minimal degree approximation (MDA) theorem and a computation algorithm. The main theorem describes a closed-form formulation for the optimal approximants, with the optimality verified by a complete error analysis. In deriving the main theorem, some useful singular value/vector properties associated with block-Hankel matrices are explored and a key extension theorem is also developed. Imbedded in the polynomial-theoretic derivation of the extension theorem is an efficient approximation algorithm. This algorithm consists of three steps: i) compute the minimal basis solution of a polynomial matrix equation; ii) solve an algebraic Riccati equation; and iii) find the partial fraction expansion of a rational matrix.

Original language | English (US) |
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Pages (from-to) | 832-852 |

Number of pages | 21 |

Journal | IEEE Transactions on Automatic Control |

Volume | 26 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1981 |

Externally published | Yes |

### All Science Journal Classification (ASJC) codes

- Control and Systems Engineering
- Computer Science Applications
- Electrical and Electronic Engineering

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## Cite this

*IEEE Transactions on Automatic Control*,

*26*(4), 832-852. https://doi.org/10.1109/TAC.1981.1102736