This paper discusses the problem of designing high gain feedback systems with robust stability.
Concerned with disturbance attenuation and/or transient response, it is known that a high gain feedback control is of great worth. However, to make a high gain feedback scheme more reliable, it is important to consider the robust stability problem against plant perturbation.
In this paper we suppose that the feedback system has a nonlinear feedback gain
k(y)/y≥1 as a high gain element, and has the SISO plant with uncertainty of the class of M(
p0, r) ={
p:
p=(1+
l)
p0, |
l(
jω)|≤|
r(jω)|, ∀ω∈R,
l does not change the number of unstable poles}, where
p0 and
l denote the nominal plant and perturbation, respectively, and are proper real rational functions.
We call a triple (
p0,
c,
r) is robustly positive real if
pc/(1+
pc) is stable and positive real for all plant in M(
p0,
r). It is obvious, from passivity theorem, that the robust positive reality of (
p0,
c,
r) guarantees the stability of the feedback system for all nonlinear gain
k(y)/y≥1 and for all perturbation in M(
p0,
r). Hence we only deal with the problem of designing a compensator which achieves robust positive realness.
The main results of this paper are essentially stated as follows. A given triple (
p0,
c,
r) is robustly positive real if and only if the following conditions hold.
1) c stabilizes
p0, 2) |
r(jω)||
a0(jω)|<1, ∀ω∈R
e and 3) |
r(jω)||
t0(jω)|+|1-
t0(jω)|≤1, ∀ω∈R, where
a0:=
p0c/(1+
p0c),
t0:=2
p0c/(1+2
p0c).
As for the realizability of compensators, we show the following result. Suppose that the roots of |
r(jω)|
2-1=0 are finitely many and are with multiplicities ≤2. Then for a given pair (
p0, r) there exists a compensator that attains robust positive realness if and only if the followings are satisfied:
1)
p0 has neither finite zeros in the open right-half plane nor, multiple
jω-axis zeros (including
j∞), 2) |
r(jω)|{=0, for
jω-axis zero of
p0, <1, for jω-axis pole of
p0, ≤1, elsewhere.
It should be noted that the modeling error must be less than or equal to 100% and, in particular, must be zero at the
jω-axis zeros of
p0 for achieving the robust positive realness.
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