Chapter 5 Classical Feedback Analysis
Xmath Control Design Module 5-2 ni.com
Because open-loop systems are generally easier to study and model than
closed-loop systems, you want to design closed-loop systems based on
information obtainable from the open-loop system.
Root Locus
In Chapter 4, System Analysis, you learned how the location of the system
poles and zeros affects the stability of the system, so an effective feedback
control design should take into account the closed-loop pole and zero
locations. If you represent the open-loop transfer function H(s) as the
quotient of the numerator and denominator as follows:
you can rewrite the characteristic equation of the closed-loop system as
follows:
This restates the fact that the open-loop system poles (which correspond
to K = 0) are the roots of the transfer function denominator, D(s). As K
becomes large, the roots of the previous characteristic equation approach
the roots of N(s)—the zeros of the open-loop system—or infinity. For a
closed-loop system with a nonzero, finite gain K, the solutions to the
preceding equation are given by the values of s where both of the following
are true:
The root locus is a plot in the real-imaginary axis showing the values
of sthat correspond to pole locations for all gains K, starting at K = 0
(the open-loop poles) and ending at K = ∞.
Root locus plots provide an important indication of what gain ranges can
be used while keeping the closed-loop system stable. As discussed in the
System Stability: Poles and Zeros section of Chapter 4, System Analysis,
continuous-time systems are stable as long as their poles are in the left half
of the s-plane (have a negative real part) and discrete-time systems are
stable as long as their poles remain within the z-plane unit circle.
The Xmath root locus-plotting utility exists for SISO systems only, though
either state-space or transfer function models can be specified.
Hs() Ns() Ds()⁄=
1KH s()+Ds() KN s()+0==
KH s() 1=∠Hs() 2k1+()π±=k01…,,=()