The theory of the asymptotic behaviour of the root-loci of linear, time-invariant, multivariable, feedback systems is developed. It is shown that each of the system zeros attracts, and is a terminating point of, one of the root-loci, as the feedback gain tends to infinity. The root-loci, that are not attracted by the zeros, tend to infinity in a special pattern that is dictated by the eigen-properties of the elementary matrices of the system. To complete the geometric description of the asymptotic behaviour of the root-loci, the concept of infinite zeros and their order is introduced. Each infinite zero of order r attracts one root-locus and, together with r−1 other infinite zeros of the same order, the corresponding asymptotes form a Butterworth configuration of order r around a special point defined as a ‘ pivot ’. A detailed algorithm for the calculation of the finite and infinite zeros is given and is illustrated by examples. A synthesis technique is then proposed by which a constant feedback controll...
[1]
W. Evans,et al.
Graphical Analysis of Control Systems
,
1948,
Transactions of the American Institute of Electrical Engineers.
[2]
Ching-Hwang Hsu,et al.
A proof of the stability of multivariable feedback systems
,
1968
.
[3]
H. Rosenbrock,et al.
State-space and multivariable theory,
,
1970
.
[4]
A. Morse,et al.
Status of noninteracting control
,
1971
.
[5]
J. J. Belletrutti,et al.
The characteristic locus design method
,
1973
.
[6]
E. Davison,et al.
Properties and calculation of transmission zeros of linear multivariable systems
,
1974,
Autom..
[7]
Charles A. Desoer,et al.
Zeros and poles of matrix transfer functions and their dynamical interpretation
,
1974
.