Stability analysis of time-delay systems via Nyquist stability criterion and multidimensional systems theory

Discusses the stability of time-delay systems described by the state vector differential-difference equations in view of the Nyquist stability criterion and multidimensional systems theory. The stability of the time-delay systems described by state vector differential-difference equations can be checked by premultiplying a real rational matrix by a diagonal time-delay matrix and plotting the eigenloci of the resulting matrix. The strictly bounded real (SBR) condition of this matrix gives conservative sufficient conditions of stability independent of delay (i.o.d.) for time-delay systems, which can be improved by introducing real rational similarity transformation matrices. The first order scalar transfer functions which are lossless bounded real independent of delay (LBR i.o.d.) always make it possible to derive real coefficients minimal realization systems; but if they are LBR dependent on delay (LBR d.o.d.) it is impossible to derive minimal realization systems without using complex coefficients systems