Arbitrary matrix coefficient assignment for block matrix linear control systems by static output feedback
Vasilii Zaitsev, Inna Kim
TL;DR
The paper addresses arbitrarily assigning the matrix coefficients of the characteristic matrix polynomial (CMP) for block-matrix linear control systems via linear static output feedback (LSOF). It develops constructive sufficient conditions for solvability when the state matrix is structured as a lower block Frobenius form or a lower block Hessenberg form, with zero blocks in input/output, and reduces AMCA to a full-rank condition on a block-structured matrix Θ, accompanied by algorithms to compute the gain and a similarity transform. The results extend earlier scalar-block findings to general block sizes, include corollaries for special cases, and connect AMCA to block pole assignment through left solvents, providing practical design tools and insights into spectral shaping of block-structured systems. The work is supported by modeling examples and algorithmic procedures, though several theoretical questions remain open, such as necessity of the rank conditions and extensions to more general block configurations.
Abstract
This work has introduced a generalized formulation of the problem of eigenvalue spectrum assignment for block matrix systems. In this problem, it is required to construct a feedback that provides that the matrix of the closed-loop system is similar to a block companion matrix with arbitrary predetermined block matrix coefficients. Sufficient conditions for the resolvability of this problem by linear static output feedback are obtained when the coefficients of the system have a special form, namely, the state matrix is a lower block Frobenius matrix or a lower block Hessenberg matrix, and the input and output block matrix coefficients contain some zero blocks. These conditions are controllability-like rank conditions. Sufficient conditions are constructive. It is proved that, in particular cases, when the system has block scalar matrix coefficients, these conditions can be weakened. The results generalize the previous results obtained for the case of one-dimensional blocks and for the case of systems given by a linear differential equation of higher orders with a multidimensional state. Based on the main results, algorithms are developed that ensure the construction of a gain matrix. The algorithms are implemented on a modeling examples.