On Nyquist Approach to Stability Analysis for Time-Delay Systems and Properties of SBR Matrices

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The stabilities of the time-delay systems described by state vector differential-difference equations are discussed in view of Nyquist stability criterion and multidimensional systems theory. First, an intermediate realization system is introduced, and the premultiplication of the leading block submatrix of this system by a block diagonal time-delay matrix is done. Stability of time-delay systems can be checked by plotting the eigenloci of the the resulting matrix. The stability independent of delay (i. o. d.) of time-delay systems can be checked by investigating the strictly bounded real (SBR) condition of this matrix instead of plotting the eigenloci of this matrix. SBR condition gives conservative sufficient conditions of stability i. o. d. for time-delay systems, which can be improved by introducing real rational similarity transformation matrices. It is also shown that the first order scalar transfer functions which are loss-less bounded real independent of delay (LBR i. o. d.) are always possible to derive real coefficients minimal realization systems. But, if the transfer functions are LBR dependent on delay (LBR d. o. d.) it is impossible to derive minimal realization systems without using complex coefficients systems.
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