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**On the bounded input-bounded output stability of a second-order linear difference equation.**
*(English)*
Zbl 0579.93053

The paper deals with the question whether or not linear time-variant difference equations are bounded output stable provided they have a bounded input sequence (BIBO stability). If the system can be described by a lower triangular system matrix, a necessary and sufficient condition for BIBO stability is easily proved using a known result of Toeplitz. This result may also be rewritten as a system of first order linear difference equations with time-variant coefficients. If the system represented by a lower triangular system matrix is decided to be BIBO stable, a necessary and sufficient condition is derived, asserting that the output sequence is converging to zero if the same holds for the input sequence.

For transformation of a system of two linear time-variant first order difference equations into lower triangular form a possible transformation matrix is chosen. With the aid of this result a linear time-variant second order difference equation is transformed into an equivalent lower triangular system of two first order equations. The necessary and sufficient stability conditions in this particular case are presented. For determination of the only time-variant element of the transformation matrix a second order difference equation is derived which turns out to be the homogeneous part of the original second order equation.

The results obtained for the general time-variant case are then specialized to time-invariant systems. The difference equation for the only element of the transformation matrix which depends on the original second order system then turns out to be the recurrent relation for finding the root with greatest absolute value. Two examples illustrate application of the presented results which are of valuable interest, particularly if partial systems of second order are only given numerically.

For transformation of a system of two linear time-variant first order difference equations into lower triangular form a possible transformation matrix is chosen. With the aid of this result a linear time-variant second order difference equation is transformed into an equivalent lower triangular system of two first order equations. The necessary and sufficient stability conditions in this particular case are presented. For determination of the only time-variant element of the transformation matrix a second order difference equation is derived which turns out to be the homogeneous part of the original second order equation.

The results obtained for the general time-variant case are then specialized to time-invariant systems. The difference equation for the only element of the transformation matrix which depends on the original second order system then turns out to be the recurrent relation for finding the root with greatest absolute value. Two examples illustrate application of the presented results which are of valuable interest, particularly if partial systems of second order are only given numerically.

Reviewer: H.-D.Fischer

### MSC:

93D25 | Input-output approaches in control theory |

93C05 | Linear systems in control theory |

93C55 | Discrete-time control/observation systems |

39A11 | Stability of difference equations (MSC2000) |

93C99 | Model systems in control theory |

93D20 | Asymptotic stability in control theory |

93B17 | Transformations |

### Keywords:

bounded-input bounded-output stability; transformation; lower triangular form; linear time-variant second order difference equation
Full Text:
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### References:

[1] | Li Ta: Die Stabilitätsfrage bei Differenzengleichungen. Acta Math. 63 (1934), 99-141. · Zbl 0009.35901 |

[2] | H. S. Wall: Analytic Theory of Continued Fractions. Van Nostrand, Toronto 1948. · Zbl 0035.03601 |

[3] | L. Prouza: Zur Stabilität der linearen Impulsfilter. Kybernetika 3 (1967), 6, 587-599. · Zbl 0155.14803 |

[4] | J. Matyáš: Untersuchung der Stabilität eines neuen diskreten Filters. Arch. El. Übertragung 38 (1984), 1, 64-68. |

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