经典文献：执行器受限处理与模型参考自适应控制
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"这篇文章是1974年发表在IEEE Transactions on Automatic Control期刊上的，标题为'1974-TAC-Model reference adaptive control with an augmented error signal'，作者未在摘要中给出。该文献主要探讨了模型参考自适应控制（Model Reference Adaptive Control, MRAC）的一个关键问题——执行器受限处理。它引入了一种增强误差信号的方法来解决这一问题，该方法在输入饱和处理方面具有经典意义，并对后来的相关研究产生了深远影响。"
本文涉及的核心知识点：
1. **模型参考自适应控制（Model Reference Adaptive Control, MRAC）**：MRAC是一种自动控制系统设计方法，其目标是使实际系统的性能尽可能接近于理想参考模型的性能。在MRAC系统中，控制器会根据系统性能与参考模型之间的偏差自我调整，以达到最佳控制效果。
2. **执行器受限（Actuator Constraints）**：在实际控制系统中，执行器的能力受到物理限制，例如电机的最大转速、液压系统的压力范围等。执行器受限是控制系统设计必须考虑的关键因素，因为它可能导致系统性能下降或损坏设备。
3. **输入饱和处理**：当控制输入超过执行器的最大允许值时，就会发生输入饱和。这种情况下，控制信号不再能直接影响系统状态，导致控制效果减弱。文献提出的增强误差信号方法旨在解决这个问题，确保即使在输入饱和的情况下，系统仍能保持稳定和良好的性能。
4. **误差动态补偿（Error Dynamic Compensation）**：这是一种策略，通过设计控制器来减少或消除由于各种原因（如模型不确定性、执行器受限等）产生的误差。在本论文中，增强的误差信号被用于动态地调整控制信号，以补偿因输入饱和造成的控制效果损失。
5. **历史文献引用**：摘要提到了几部在控制理论和优化方法领域具有里程碑意义的书籍，如J.A. Bryson和Y.C. Ho的《Applied Optimal Control》、J.B. Meditch的《Stochastic Optimal Linear Estimation and Control》等，这些书籍为理解本文的研究背景提供了参考。
6. **学习与优化**：文献可能也讨论了如何通过学习算法和优化技术来改进控制器的设计，以适应系统参数的变化和执行器的限制。这可能涉及到如强化学习等机器学习方法，以及如R.L. Fox的《Optimization Methods for Engineering Design》中提到的工程优化技术。
该文献是输入受限问题处理的重要先驱，其增强误差信号的处理方法在自适应控制领域具有重要地位，对于理解和解决实际系统中的控制挑战有着重要的学术价值和实践意义。
476
IEEE
TRANSACTIONS
ON
AUTO~IATIC
CONTROL,
OCTOBER
1974
the application
of
the
MKY
lemma to these systevs.
The
problem solut,ion for a limited class of plants
is
discussed
in Section
IV,
and for the general class in Sect.ion
V.
A
computer simulation is included in Section
VI,
and the
conclusions in Section JII.
11.
PROBLEM
STATEMENT
AXD
DESIGN
OBJECTIVES
A
dynamic system (plant)
is
described by t,he non-
linear, nonautomonous, different,ial equat.ion
D,(P)
40
=
WP)
40
+
cf(x,O (1)
where
u(t)
and
x(t)
are the plant input and output,
re-
spect,ively,
f(~$)
is
a nonlinear time varying funct>ion of
known form,
p
is
the operat.or
d/dt,
D&)
=
p"
+
ulpn--l
+
. .
a,,
and
D,(p)
=
boprn
+
blpm-'
+
.
. .
b,.
It
is
assumed that coefficients
a,
bi,
and
c
are unknown and
constant,
or
slowly varying. The term
cf(x,t)
may be
re-
placed by
a
sum of terms of the same type. Since each
is
handled the same way in the design, there is no loss of
generality
in
carrying only the one term.
It.
is
also assumed
that
1)
the function
f
(after first being introduced, the
arguments of funct,ions and time variables will be omit,ted,
with t,he exception of
p
operators) satisfies t,he continuity
condit,ions necessary for solut.ions of
(1)
to exist and be
unique,
2)
all roots of
D,(s)
are in the open left half plane,
3)
bo
#
0,
and
4)
m
5
n
-
1.
Assumption
2)
is
required to insure a bounded control
input by not requiring the a.daptive syst.em t,o act
to
cancel
nonminimum phase plant zeroes. For a similar reason,
u
is
not included as
an
argument
off.
In certain cases func-
tions
of
t.he form
f(z,u,t)
may
be
allowed.
However,
in
general, it
is
not. known what rest.rictions to impose on
these functions to insure that- the cont.ro1 input
u
is
bounded. In the purely t,heoretical sense assumpt,ion
3)
is
not necessary. The system will adapt and follow t,he model
but the control input will become unbounded-an unde-
sirable practical result.. Similarly
f(x,u,t)
can be allowed
theoret.ically provided it does not, impose a magnitude
constraint, on
u.
The design objective is to have the plant out.put follow
the output of
a
model reference defined by the equation
D,(P)
s,(t>
=
KO
r(i>
+
g(x,,r7f) (2)
where
D,(p)
=
pn
+
udl
pX-l
+
. .
.
ad,,
x,
is
the model
output,
r
is
t,he reference input
x,
is
the model state vector,
and
g
is a nonlinear, time varying function wit.h t,he
smoot,hness properties required to insure the existence and
uniqueness of solut,ions to
(2).
If
it is considered desirable
t.0
have a model which differentiates the reference input,
~(t)
may be replaced by
r(t)
=
D,(p)rl(t)
where
D,(p)
=
pa
+
RIP"-'
+
. .
.
R,.
If
a
_<
m,
no derivatives of
,rl
will be required in the design. However,
if
a
>
m,
t.hep
will
be. Stability of the operator
Dm@)
is
assumed, i.e.,
D,(s)
=
0
has roots in the open left half plane only. The
design problem
is
to synthesize a parameter adaptive
control system for
(1)
which will muse the error,
e(t)
=
x,
-
x,
between plant and model outputs
t.0
approach
zero. The distinguishing feature between this work and
previous designs of this type is that derivat.ives of the
plant, output,
x
are not, to be used in the design for the
obvious reason that. in pract>ice they may be t.oo noisy.
A
new concept, an augment,ed error signal, must be
introduced tmo achieve the design objective.
This
signal
used
in
conjunction wit.h the state variable filter concept
makes it possible
to
derive syst.em differential equations
in
a
form suitable for direct application of the
3IKY
lemma, and
a,s
in ot.her designs using t,his lemma
[4],
1111,
[16],
no derivatives of
z
are required.
An
augmenting signal
y(f)
is
added to
e@)
t,o give the
augmented
error
signal
~(t),
do
=
4)
+
Y(0.
(3)
The signal
~(t)
is
the output. of the error augmenting filter
defined by
D,(P)Y
=
D,(P)U.(O
(4
where
D,(p)
=
pn-'
+
~~p~-~
+
.
. .
c,,
and
'ZL'
is
an
auxiliary system input to be determined along wit.h the
control input
u
as
part of the design. C0efficient.s
ci
in
D,(p)
must be chosen such that, t,he transfer funct.ion
D,(s)/D,(s)
is
a
posit.ive real function of
s
in order to use
the
3IBY
lemma lat.er in the development.
Subtracting
(1)
from
(2)
gives the different,ial equation
for
e
which
is
D,(p)e
=
&r
+
g
-
D,@)u
-
cf
+
D,(P)z
(5)
where
D,(p)
=
D,(p)
-
Dm(p)
=
Actl
pn-'
+
.
. .
Au,
and
Aai
=
ai
-
udi
for
i
=
1
to
7%.
Xext,
(4)
is
added
to
(3)
to
obt.ain thv differential
equation for the augmented error signal
7,
which
is
D,(p)s
=
Kor
-I-
g
-
D,(p)u
-
cf
+
DA(p)x
+
D,(p)-zc.
(6)
Equat,ion
(6)
is
the starting equation for the synt,hesis
procedure. The design problem
is
to generat.e
u
and
w
independent of derivat.ives of
x
and in such a way that
e
0
as
1
---f
.
In a sense, the augmenting signal
y
is
a
catalyst which plays
a
role only during t,he adapt,ive
transient period and goes
to
zero once adaptat,ion
is
com-
plet.ed.
Its
successful applicat.ion in this problem leads one
to
speculate about, the possibilit*y
of
using the concept of
an augmented error signal to insure convergence in other
types
of
gradient algorit,hms.
111.
THE
RII<\-
LEMMA
IN
ADAPTIVE COSTROL
SYSTEM DESIGN
A
statement of the
MI<Y
lemma
is
given in Bppendix I.
The reason for it,s usefulness in designs of this type
is
ex-
plored in this sect.ion.
Consider t,he system
pe(t)
=
Ae(l)
+
dr(t)
(7)
with output
e
=
k'
e.
Here
A,
d,
and
k
are the
71.
X
12
ma.trix and
n
vectors referred to in the lemma. The lemma
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