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State-Space Approach to Control System Design
Published in Raymond G. Jacquot, Modern Digital Control Systems, 2019
For the polynomial matrix () αc(A1)=αc(T−1AT)=T−1αc(A)T
Matrices and Linear Algebra
Published in William S. Levine, Control System Fundamentals, 2019
For equivalence of polynomial matrices (see [3] for details), where R = ℝ[s] (or ℂ[s]), let P(s) and Q(s) be invertible n × n polynomial matrices. Such matrices are called unimodular; they have constant, nonzero determinants. Let A(s) be an n × n polynomial matrix with nonzero determinant. Then, for the equivalent matrix A¯(s)=P(s)A(s)Q(s),det A¯(s) differs from det A(s) only by a constant factor; with no loss of generality, A¯(s) may be assumed to be scaled so that det A¯(s) is a monic polynomial, i.e., so that the coefficient of the highest power of s in det A¯(s) is 1.
Matrices and Linear Algebra
Published in William S. Levine, The Control Handbook: Control System Fundamentals, 2017
For equivalence of polynomial matrices (see [5] for details), where R = ℝ[s] (or ℂ[s]), let P(s) and Q(s) be invertible n × n polynomial matrices. Such matrices are called unimodular; they have constant, nonzero determinants. Let A(s)bean n × n polynomial matrix with nonzero determinant. Then for the equivalent matrix A¯(s)=P(s)A(s)Q, det A¯(s) differs from det A(s) only by a constant factor; with no loss of generality A¯(s) may be assumed to be scaled so that det A¯(s) is a monic polynomial, that is, so that the coefficient of the highest power of s in det A¯(s) is 1.
Degree of Dieudonné determinant defines the order of nonlinear system
Published in International Journal of Control, 2019
Ü. Kotta, J. Belikov, M. Halás, A. Leibak
The degree of the determinant has been used in linear control theory in many different purposes. First, it reveals the system order (Kailath, 1980, p. 369), the number of its poles and zeros, and can additionally be used to investigate both the finite and infinite structure of system (Henrion & Šebek, 1999). Some results have been formulated with the help of this concept. For instance, the realisation of is minimal only if the degree of the determinant of is equal to the degree of the rational matrix ; see Chen (1970). Next, the degree of the determinant of the so-called interactor matrix helps to determine the necessary and sufficient conditions for MIMO linear system to be prime (Baser & Eldem, 1984). In linear control, one frequently needs to compute numerically the determinant, inverse or Smith form of a bivariate polynomial matrix using evaluation--interpolation techniques. The number of interpolation points depends on the degree of determinant. The paper Varsamis and Karampetakis (2014) presents a recursive formula to find the degree of the determinant of a bivariate polynomial matrix. For earlier results on univariate case, see Henrion and Šebek (1999), and Hromčík and Šebek (1999).