Generalized selection of complementary matrices in the inclusion principle.

*(English)*Zbl 0990.93002A fundamental fact in the so-called “Inclusion Principle” introduced in the early 1980’s by M. Ikeda and D. D. Siljak [Overlapping decompositions, expansions and contractions of dynamic systems, Large Scale Syst. 1, 29-38 (1980; Zbl 0443.93009)] and concerning the analysis and control of complex and large scale systems, relies on an appropriate selection of the “complementary matrices” appearing in the linear transformations between the inputs, states and outputs of dynamic systems with different dimensions in which solutions of the system with larger dimension includes solutions of the system with smaller dimension.

As a continuation of previous works of the first two authors [Int. J. Control 61, 559-570, 571-587 (1995; Zbl 0825.93025 and Zbl 0825.93026)] and of the authors [SIAM J. Matrix Anal. Appl. 21, No. 4, 1136-1155 (2000; Zbl 0979.93006)], a strategy is proposed for choosing the complementary matrices when dealing with state LQ optimal control for linear time-invariant systems including the contractibility conditions. The main results (Section III) are concerned with identifying a new block structure of the complementary matrices that generalize well-known results for expansion-contraction of pairs of systems and optimal control criteria. Some illustrative numerical results are also given.

As a continuation of previous works of the first two authors [Int. J. Control 61, 559-570, 571-587 (1995; Zbl 0825.93025 and Zbl 0825.93026)] and of the authors [SIAM J. Matrix Anal. Appl. 21, No. 4, 1136-1155 (2000; Zbl 0979.93006)], a strategy is proposed for choosing the complementary matrices when dealing with state LQ optimal control for linear time-invariant systems including the contractibility conditions. The main results (Section III) are concerned with identifying a new block structure of the complementary matrices that generalize well-known results for expansion-contraction of pairs of systems and optimal control criteria. Some illustrative numerical results are also given.

Reviewer: Pablo Gonzales-Vera (La Laguna)

##### MSC:

93A14 | Decentralized systems |

93A15 | Large-scale systems |

49N10 | Linear-quadratic optimal control problems |

93B11 | System structure simplification |

15A04 | Linear transformations, semilinear transformations |

93B17 | Transformations |