状态反馈控制：延缓指数RED算法的Hopf分岔
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本文探讨了在Hopf分岔情况下，状态反馈控制在一种新型网络拥塞控制模型——指数随机早期丢弃算法（Exponential Random Early Detection, RED）中的应用。该模型考虑了一条链路和单个源的情况。研究者Min Xiao、Guoping Jiang 和 Lindu Zhao 在 Nonlinear Dyn 杂志(2014年卷76, 第1469-1484页)中提出了他们的研究成果，该论文的DOI为10.1007/s11071-013-1221-0。
在传统的网络拥塞控制中，状态反馈控制方法常用于稳定不稳定稳态或周期性行为。然而，对于这种指数RED算法，当不采用控制时，模型会在一个特定的增益参数区间内发生分岔，导致发送速率的不稳定。这一分岔可能会阻碍系统的正常性能，使得网络在增益参数的其他区间无法维持稳定的发送速率。
作者引入状态反馈控制作为解决方案，将增益参数视为控制参数。通过精心设计的控制器，他们成功地推迟了Hopf分岔的发生，即使在分岔区域之外，也能够保证网络在一定的增益范围内维持稳定的发送速率。这表明状态反馈控制不仅能够稳定已知的不稳定状态，还能有效地管理在网络运行过程中可能出现的复杂动态行为，如Hopf分岔。
文中还提供了数值模拟来验证状态反馈控制器的有效性。这些模拟展示了在实施控制后，系统如何在预期的范围内保持稳定，避免了因Hopf分岔带来的网络性能下降。这项研究不仅对理论分析有所贡献，也为实际网络拥塞控制策略的设计提供了实用的指导，强调了在面对网络动态变化时，适时且精准的状态反馈控制的重要性。
1472 M. Xiao et al.
˙x(t) = x(t − τ)
1− p(t)
τ
2
x(t)
− kx(t) p(t)
− α
1
(x(t) − x
∗
) − α
2
(x(t) − x
∗
)
2
− α
3
(x(t) − x
∗
)
3
,
˙p(t) =
β
c
p(t)(x(t − τ)− c).
(8)
3.1 Stability and existence of bifurcation
for controlled system (8)
Let u
1
(t) = x(t) − x
∗
, u
2
(t) = p(t) − p
∗
. Then, (8)
becomes
˙u
1
(t) = (u
1
(t − τ) + c)
kc
2
τ
2
−(1+kc
2
τ
2
)u
2
(t)
τ
2
(1+kc
2
τ
2
)(u
1
(t)+c)
− k(u
1
(t) + c)(u
2
(t) +
1
1+kc
2
τ
2
)
− α
1
u
1
(t) − α
2
u
2
1
(t) − α
3
u
3
1
(t),
˙u
2
(t) =
β
c
u
2
(t) +
1
1+kc
2
τ
2
u
1
(t − τ).
(9)
Linearizing (9) about (0, 0) produces
˙u
1
(t) =
−
2kc
1+kc
2
τ
2
−α
1
u
1
(t)−
1+kc
2
τ
2
τ
2
u
2
(t),
˙u
2
(t) =
β
c(1+kc
2
τ
2
)
u
1
(t − τ),
(10)
which has the characteristic equation:
λ
2
+
2kc
1 + kc
2
τ
2
+ α
1
λ +
β
cτ
2
e
−λτ
= 0. (11)
In what follows, we regard β as the bifurcation para-
meter to investigate the distribution of the roots to (11).
Lemma 1 If α
1
> 0, then there exists a minimum pos-
itive number β
c
0
such that
(i) (11) has a pair of purely imaginary roots ±iω
c
0
when β = β
c
0
.
(ii) all the roots of (11) have negative real parts when
β ∈ (0,β
c
0
).
(iii) β
c
0
>β
0
, where β
0
is defined by (3)–(5).
Here,
β
c
0
= cτ
2
(ω
c
0
)
4
+
2kc
1+kc
2
τ
2
+α
1
2
(ω
c
0
)
2
, (12)
and
ω
c
0
∈
0,
π
2τ
(13)
is the root of the equation
1
2kc
1+kc
2
τ
2
+ α
1
ω = cot(ωτ ). (14)
Proof (i) If λ = iω (ω>0) is a pure imaginary solu-
tion of (11), it is straightforward to obtain that
ω
2
=
β
cτ
2
cos(ωτ ),
2kc
1+kc
2
τ
2
+ α
1
ω =
β
cτ
2
sin(ωτ ).
(15)
Taking the ratio of the two equations of (15) yields
(14). Solutions of (14) are the horizontal coordinates of
the intersecting points between the curve y = cot(ωτ )
and the line y =
1
(
2kc
1+kc
2
τ
2
+α
1
)τ
ωτ. There are infinite
numbers of intersecting points for these two curves that
are graphically illustrated in Fig. 2.
Let ω
c
0
satisfy (13) and be the root of (14) and define
β
c
0
as in (12). Then, (β
c
0
,ω
c
0
) is a solution of (15). Thus,
±iω
c
0
is a pair of purely imaginary roots of (11) when
β = β
c
0
. It is easily seen form Fig. 2 that τω
c
0
is the
minimum positive value among all horizontal coordi-
nates of the intersecting points. So, β
c
0
is the first value
of β>0 such that (11) has root appearing on the imag-
inary axis. The conclusion (i) follows.
(ii) When β = 0, the root of (11)is
λ
1
=−
2kc
1 + kc
2
τ
2
+ α
1
< 0,λ
2
= 0.
Let λ
2
(β) be a root of (11) satisfying λ
2
(0) = 0. We
can obtain that
λ
2
(0) =−
1
2kc
1+kc
2
τ
2
+ α
1
cτ
2
< 0.
Thus, all roots of (11) have negative real parts when
β ∈ (0,β
c
0
). The conclusion (ii) follows.
(iii) It is clear from Fig. 2 that τω
0
is the horizon-
tal coordinate of the intersecting point between the
curve y = cot(ωτ ) and the line y =
1
2kc
1+kc
2
τ
2
τ
ωτ,
while τω
c
0
is the horizontal coordinate of the intersect-
ing point between the curve y = cot(ωτ ) and the line
y =
1
(
2kc
1+kc
2
τ
2
+α
1
)τ
ωτ.
If α
1
> 0 holds, we have
1
2kc
1+kc
2
τ
2
τ
>
1
(
2kc
1+kc
2
τ
2
+α
1
)τ
.
Therefore, τω
c
0
>τω
0
, i.e., ω
c
0
>ω
0
. From the defini-
tions of β
0
and β
c
0
in (3) and (12), respectively, we can
obtain that β
c
0
is larger than β
0
. The conclusion (iii)
follows. Then the proof is completed.
Remark 6 If the controller u is removed from the con-
trolled system (8), i.e., α
1
= α
2
= α
3
= 0, then ( 12)
can be identical with the expression of β
0
in Theorem 1.
Therefore, β
0
in Theorem 1 is a special case of β
c
0
in
(12) in the absence of the control.
123
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