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论文研究 - 一类非线性系统输出反馈分布式模型预测控制器的设计.pdf
The Design of Output Feedback Distributed Model Predictive Controller for a Class of Nonlinear Systems
Abstract
Keywords
1. Introduction
2. Preliminaries
2.1. Definitions and Lemmas
2.2. Problem Formulation
3. Observers and Property
3.1. The Design of Observers
3.2. The Property of Observers
4. Lyapunov-Based Controller
5. Output Feedback Distributed Model Predictive Control
5.1. Distributed Model Predictive Control
5.2. Stability Analysis
6. Example
7. Conclusion
Acknowledgements
References
Applied Mathematics, 2017, 8, 1832-1850
http://www.scirp.org/journal/am
ISSN Online: 2152-7393
ISSN Print: 2152-7385
The Design of Output Feedback Distributed
Model Predictive Controller for a Class of
Nonlinear Systems
Baili Su, Yingzhi Wang
College of Engineering, Qufu Normal University, Rizhao, China
How to cite this paper: Su, B.L. and Wang,
Y.Z. (2017) The Design of Output Feedback
Distributed Model Predictive Controller for
a Class of Nonlinear Systems. Applied
Mathematics, 8, 1832-1850.
https://doi.org/10.4236/am.2017.812131
Received: December 1, 2017
Accepted: December 26, 2017
Published: December 29, 2017
Copyright © 2017 by authors and
Scientific Research Publishing Inc.
This work is licensed under the Creative
Commons Attribution International
License (CC BY 4.0).
http://creativecommons.org/licenses/by/4.0/
Open Access
Abstract
For a class of nonlinear systems whose states are immeasurable, when the
outputs of the system are sampled asynchronously, by introducing a state ob-
server, an output feedback distributed model predictive control algorithm is
proposed. It is proved that the errors of estimated states and the actual sys-
tem's states are bounded. And it is guaranteed that the estimated states of the
closed-loop system are ultimately bounded in a region containing the origin.
As a result, the states of the actual system are ultimately bounded. A simula-
tion example verifies the effectiveness of the proposed distributed control
method.
Keywords
Nonlinear Systems, Distributed Model Predictive Control, State Observer,
Output Feedback, Asynchronous Measurements
1. Introduction
Traditional process control systems just simply combine the measurement sen-
sors with control actuators to ensure the stability of closed-loop systems. Al-
though this paradigm to process control has been successful, the calculation
burden of this kind of control is large and the performance of the system is not
good enough [1]. So far the stability of closed-loop systems has been guaranteed
and at the same time, the performance of the closed-loop systems has been im-
proved if the control systems are divided into local control systems (LCS) and
networked control systems (NCS). And it can reduce the burden of calculation.
But this kind of transformation needs to redesign LCS and NCS to ensure the
stability of closed-loop systems. As a result, the control strategy is changed [2].
DOI: 10.4236/am.2017.812131 Dec. 29, 2017
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B. L. Su, Y. Z. Wang
Model predictive control (MPC) is receding horizon control which can deal
with the constraints of systems’ inputs and states during the design of optimiza-
tion control. It adopts feedback correction, rolling optimization, and has strong
ability to deal with constraints and dynamic performance [3] [4] [5]. Therefore,
it can be more effective to solve the optimal control problem for distributed sys-
tems. That is distributed model predictive control [6]. The distributed model
predictive control takes into account the actions of the local controller in the
calculation of its optimal input trajectories. At the same time, LCS and NCS are
designed via Lyapunov-based model predictive control (LMPC). But when the
LCS is a model predictive control system for which there is no explicit control
formula to complete its future control actions, it is necessary to redesign both
the NCS and LCS and establish some communication between them so that they
2u as
can coordinate their actions. We refer to the trajectories of
LMPC1 and LMPC2. The structure of the system is as Figure 1.
1u and
There are many research results about distributed MPC design at present. In
literature [7], a novel partition method of distributed model predictive control
for a class of large-scale systems is presented. Literature [8] presents a coopera-
tive distributed model predictive control algorithm for a team of linear subsys-
tems with the coupled cost and coupled constraints. A distributed model predic-
tive control architecture of nonlinear systems is studied in literature [2]. Based
on literature [2], literature [9] considers a distributed model predictive control
method subject to asynchronous and delayed measurements. A distributed
model predictive control strategy for interconnected process systems is proposed
in reference [10]. In literature [11], a design approach of robust distributed
model predictive control is proposed for polytopic uncertain networked control
systems with time delays. Reference [12] presents that the distributed model
predictive control method is applied for an accurate model of an irrigation canal.
For a hybrid system that comprises wind and photovoltaic generation subsys-
tems, a battery bank and an ac load, a distributed model predictive control me-
thod is designed to ensure the closed-loop system stable in reference [13].
These references are obtained on the assumption that the systems’ states can
be measured continuously. The systems whose states are immeasurable are not
taken into account in these references. However, immeasurable states often
DOI: 10.4236/am.2017.812131
Figure 1. Distributed LMPC control architecture.
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B. L. Su, Y. Z. Wang
happen in practice. In literature [14], under the condition that the states are not
measured, a distributed model predictive control algorithm for interconnected
systems based on neighbor-to-neighbor communication is presented. Literature
[15] considers the design of robust output feedback distributed model predictive
control when the dynamics and measurements of systems are affected by
bounded noise. But both literatures are studied for the linear systems. An out-
put-feedback approach for nonlinear model predictive control with moving ho-
rizon state estimation is proposed in reference [16]. Reference [17] considers
output feedback model predictive control of stochastic nonlinear systems. Yet
these two references are centralized model predictive control methods. The
computational complexity grows significantly.
On the basis of the above references, this paper considers a class of nonlinear
systems whose states are immeasurable. By introducing a state observer, and us-
ing output feedback, under the assumption that the outputs of the system are
sampled of asynchronous measurements, an output feedback distributed model
predictive control algorithm is designed. Therefore, the ultimately boundedness
of the estimated states and the boundedness of the error between estimated
states and the actual system’s states are proved, and then it is proved that the
states of the actual system are ultimately bounded. And the stability of the
closed-loop system is guaranteed. The performance of the system is improved
and the burden of calculation is reduced.
This paper is arranged as follows. The second section is the preparation work.
In the third section, the state observer is designed, and its stability is analyzed.
The fourth section designs a controller based on Lyapunov function to make
sure the asymptotic stability of the nominal observer. In the fifth section, an
output feedback distributed model predictive control algorithm is proposed and
the stability of the closed-loop system is proved. The instance simulation is pro-
vided in the sixth section. Conclusion is given in Section 7.
2. Preliminaries
2.1. Definitions and Lemmas
In this paper, the operator
( )
x t
r
Ω = ∈
≤
r
rentiable function and
lemmas used in this paper are as follows:
denotes the derivative of
{
x R
( )
V x
:xn
⋅ denotes Euclidean norm of variates. The symbol
rΩ denotes the set
}
where V is a scalar positive definite, continuous diffe-
( )0
= and r is a positive constant. Definitions and
V
. The symbol
( )
x t
0
Definition 1 [18]: A function
ists a constant
in a given region of x and
xL such that
( )
f x is said to be locally Lipschitz if there ex-
)
(
f x
−
2x
1
for all
xL is the associated Lipschitz constant.
1x and
(
f x
2
L x
x
1
x
2
≤
−
)
Definition 2 [18]: A continuous function
( )0 = 0
γ
if it is strictly increasing and
said to belong to class if, for fixed s,
respect to r and, for fixed r,
→ ∞ belongs to class
is
belongs to class with
is decreasing with respect to s and
. A continuous function
),r sβ
),r sβ
),r sβ
0,
(
(
(
[
aγ
: 0,
)
[
)
DOI: 10.4236/am.2017.812131
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B. L. Su, Y. Z. Wang
f
0
f
n
→ as
]T
[
: 0,
(
),
r sβ
s → .
0
Lemma 1 [18]: Let [
,0 be an equilibrium point for the nonlinear sys-
0,
)
(
x
t x
=
,
tem
where
∞ × → is continuous differentiable,
}
{
D
x R x
where r is a positive constant and the Jacobian matrix
= ∈
<
]
[
x
f
∂ ∂ is bounded on D, uniformly in t. Let β be a class function and
0r be a positive constant such that
. As-
r
sume that the trajectory of the system satisfies
)
∈
∀ ≥ ≥ (1)
{
= ∈
< . Let
x R x
( )
x t
0,0r
(
x t
(
x t
(
β
D
0
}
D
∀
r
0
R
−
<
0
)
r
)
)
t
t
t
t
,
,
,
n
n
D
0
(
β≤
)
0
0
0
0
Then, there is a continuously differentiable function
V
[
: 0,
satisfies the inequalities
∞ × → that
R
D
0
)
≤
α
2
≤ −
(
α
3
)
x
x
(
)
(2)
x
+
)
V t x
,
≤
V f
∂
x
∂
(
(
t x
,
)
)
≤
α
4
(
x
)
(
α
1
V
∂
t
∂
V
∂
x
∂
,α α α and
1
where
tem is autonomous, V can be chosen independent of t.
4α are class functions defined on [
,
2
3
]00,r
. If the sys-
0,
]T
x
=
Lemma 2 [18]: Let [
,0 be an equilibrium point for the nonlinear sys-
tem
. The equilibrium point is uniformly asymptotically stable if and
only if there exist a class function β and a positive constant c, indepen-
dent of
t x
,
(
)
f
(
β≤
(
x t
0
)
,
t
−
t
0
)
,
∀ ≥ ≥ ∀
0,
t
t
0
(
x t
0
)
< (3)
c
0t , such that
( )
x t
2.2. Problem Formulation
Consider a class of nonlinear systems described as follows:
)
( )
,
( )
x t
( )
y t
=
=
(
( )
f x t u t u t w t
( )
h x
( )
,
1
( )
tυ
( )
+
,
2
(4)
2
xn
( )
x t
un
R∈
denotes the state vector which is immeasurable.
R∈
are control inputs.
R U
where
,
( )
u t
2u are restricted to be in two non-
2
( )
empty convex sets
denotes the disturbance
w t
vector.
is a measurement
R∈
noise vector. The disturbance vector and noise vector are bounded such as
w∈ , v ∈ where
is the measured output and
1u and
.
⊆
wn
( )
v t
( )
y t
R∈
R∈
R∈
n
,u
1
U
⊆
n
u
2
R
yn
vn
1
2
( )
u t
1
un
1
{
= ∈
{
v R
= ∈
w R
n
v
n
w
:
:
v
≤
w c c
,
1
>
1
c c
,
2
2
≤
>
}
0
}
0
(5)
1c and
2c are known positive real numbers. We assume that f and h are
with
= . This means that
locally Lipschitz vector functions and
the origin is an equilibrium point for system (4). And we assume that the output
0,0,0,0
h=
0,
( )
0
0
(
)
f
DOI: 10.4236/am.2017.812131
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B. L. Su, Y. Z. Wang
kt
t
0
0t
k
,
k
}0
0,1,
such that
= + ∆ = with
of system (4), y, is sampled asynchronously and measured time is denoted by
{
being the initial time, ∆
kt ≥
being a fixed time interval. Generally, there exists a possibility of arbitrarily large
periods of time in which the output cannot be measured, then the stability prop-
erties of the system is not guaranteed. In order to study the stability properties in
mT
a deterministic framework, we assume that there exists an upper bound
interval between two successive measured outputs such that
on the
{
. This assumption is reasonable from a process control pers-
t
t
max k
pective.
+ −
1
≤
T
m
}
k
Remark 1: Generally, distributed control systems are formulated on account
of the controlled systems being decoupled or partially decoupled. However, we
consider a seriously coupled process model with two sets of control inputs. This
is a common phenomenon in process control.
The objective of this paper is to propose an output feedback control architec-
ture using a state observer when the states are immeasurable. The state observer
has the potential to maintain the closed-loop stability and improve the
2u . The
closed-loop performance. We design two LMPCs to compute
structure of the system is as follows:
1u and
Remark 2: The procedure of the system shown in Figure 2 is as follows
1) When the states are immeasurable, the observer is used to estimate the
current state x.
2) LMPC2 computes the optimal input trajectory of
2u based on the esti-
mated state ˆx and sends the optimal input trajectory to process and LMPC1.
3) Once LMPC1 receives the optimal input trajectory of
2u , it evaluates the
1u based on ˆx and the optimal input trajectory of
optimal input trajectory of
2u .
4) LMPC1 sends the optimal input trajectory to process.
5) At next time, return step (1).
3. Observers and Property
3.1. The Design of Observers
Define the nominal system of system (4) as following:
DOI: 10.4236/am.2017.812131
Figure 2. Distributed LMPC architecture where the states are immeasurable.
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Applied Mathematics
( )
*
x t
( )
*
y t
=
=
*
,
( )
(
( )
f x t u t u t
(
h x
( )
)
,
1
2
*
B. L. Su, Y. Z. Wang
,0
)
(6)
*
xn
x
R∈
where
noise free output.
denotes the state vector of nominal systems,
*
y
R∈
yn
is the
Assume that there exists a deterministic nonlinear observer for the nominal
system (6):
=
ˆ
*
x
(
( )
F x t u t u t
( )
( )
ˆ
,
,
*
1
2
( )
*
y t
,
)
(7)
, where
indicates the state vector of nominal observer. From Lemma 2, there
*x for all the states
*
*ˆ,
x x
R∈
xn
xn
R∈
*ˆx asymptotically converges
such that
*ˆ
x
exists a class function β such that:
(
*
x t
( )
ˆ
*
x t
( )
*
x t
−
≤
(
β
)
−
(
ˆ
*
x t
0
)
0
,
t
−
t
0
)
(8)
We assume that F is a locally Lipschitz vector function. Note that the conver-
gence property of observer (7) is obtained based on nominal system (6) with
continuous measured output.
From the Lipschitz property of f and Definition 1, there exists a positive con-
stant
1M such that:
for all
*
x
R∈
xn
.
(
f x u u
,
,
*
1
)
,0
2
M≤
1
(9)
The actual observer of the system is obtained when the deterministic observer
is applied to system (4). The observer of system (4) is described as follows with
the state disturbance and measurement noise:
=
( )
ˆ
x F x t u t u t
,
(10)
(
y t
( )
( )
)
ˆ
,
,
(
k
1
2
where
(
)ky t
is the actual sampled measurement at
t∀ ≥ .
t
k
)
kt , for
3.2. The Property of Observers
In this subsection, the error between the actual system's states and estimated
states will be studied under the condition of state disturbance and measurement
noise when observer (10) is applied to system (4).
Theorem 1: Consider observer (10) with output measurement
(
ˆ
kx t
)
, the error of estimated state
(
)ky t
( )
ˆx t
starting
and actual
from the initial condition
state
( )
x t
is bounded:
( )
( )
e t
x t
=
−
( )
ˆ
x t
≤
for
t∀ ≥ where
t
k
(
e t
k
k
)
=
(
x t
( )
δ τ
1
=
( )
δ τ
2
=
k
k
)
(
e t
(
β
(
ˆ
x t
(
)
−
l c
2 1
l
1
q bM N
2
q
1
l
τ
1
−
e
(
1
)
,
t
−
t
k
)
+
δ
1
(
t
−
t
k
)
+
δ
2
(
t
−
t
k
)
(11)
is the initial error of the states, and
)
1
(12)
∆ +
c
2
)(
e
q
τ
1
)
1
−
DOI: 10.4236/am.2017.812131
where
1l and
2l ,
1q and
2q , b are Lipschitz constants associated with f, F
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B. L. Su, Y. Z. Wang
and h, respectively, and N is the predictive horizon.
)
,
t∀ ≥ , from (8) and
(
ˆ
*
x t
(
ˆ
x t
=
)
t
k
k
k
Proof: For
tained that:
(
*
x t
k
)
=
(
x t
k
)
, it can be ob-
( )
*
x t
−
( )
ˆ
*
x t
(
β
(
β
(
β
≤
=
=
(
*
x t
(
x t
(
e t
k
k
k
)
)
−
)
−
,
t
k
(
ˆ
*
x t
(
)
ˆ
x t
)
t
−
k
k
)
)
t
,
,
t
−
−
t
k
k
t
)
(13)
Based on the Lipschitz property of f and Definition 1, there exist constants
2,l
l
1
, such that:
( )
*
x t
( )
x t
−
=
≤
,
,
( )
1
( )
*
x t
(
( )
( )
( )
f x t u t u t w t
,
2
( )
( )
l x t
l w t
−
1
2
)
(that is to say
(
x t
+
k
)
−
(
( )
f x t u t u t
( )
( )
,
,
*
1
2
,0
)
(14)
c≤ ,
1
kt
Because of
(
*
x t
k
)
=
(
*
x t
)
−
(
x t
k
)
k
= ), and
0
( )
w t
the following inequality can be got by integrating the above inequality from
to t :
( )
x t
−
( )
*
x t
≤
(
e
l c
2 1
l
1
(
l
1
t
t
−
k
)
)
1
− =
δ
1
(
t
−
t
k
)
(15)
From the triangle inequality and inequalities (13), (15), it can be written as:
( )
x t
−
( )
ˆ
*
x t
≤
≤
( )
x t
−
(
(
e t
β
k
( )
*
x t
)
t
,
−
+
t
k
( )
*
x t
−
)
(
δ
1
+
t
( )
ˆ
*
x t
)
,
t
−
k
(16)
∀ ≥
t
t
k
From the Lipschitz property of F and Definition 1, there exist constants
2,q q
1
satisfying the following inequality:
*
*
,
,
,
ˆ
−
≤
=
( )
ˆ
*
x t
( )
ˆ
x t
( )
( )
1
( )
( )
ˆ
x t
q x t
−
1
( )
*
y t
=
t∀ ≥ . Note that
(
( )
*
y t
y t
−
(
( )
( )
*
F x t u t u t
y t
2
( )
ˆ
q y t
+
−
2
(
(
( )
h x t
y t
)
(
)
( )
h x t
≤
)
−
(
y t
)
k
−
(
( )
F x t u t u t
)
(
(
h x t
(
)
(
h x t
(
tυ
(
tυ
)
)
+
( )
( )
,
*
k
)
=
+
)
)
ˆ
t
,
,
*
1
2
*
k
k
k
k
k
k
for
,
(
y t
k
)
)
(17)
)
, hence:
(18)
Due to the Lipschitz property of h and Definition 1, there exists a constant b
such that:
Because of
(
*
x t
k
)
=
From (9) and the dynamics of
( )
*
x t
From (20) and (21), we can get:
(
y t
( )
*
y t
−
k
*
)
−
≤
−
(
x t
(
y t
( )
b x t
( )
*
y t
(
)
and the boundedness of υ, we can get:
x t
k
( )
*
y t
( )
b x t
(
tυ
−
−
+
)
)
)
)
*
k
k
k
+
≤
(
*
x t
(
y t
c
k
2
*x , it can be derived that:
−
(
M t
(
*
x t
≤
−
)
)
t
1
k
k
(19)
(20)
(21)
)
k
≤
(
bM t
1
−
t
k
)
+
c
2
(22)
DOI: 10.4236/am.2017.812131
From (17) and (22) and
t
−
t
k
≤ ∆ , it can be obtained that:
N
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Applied Mathematics
B. L. Su, Y. Z. Wang
( )
ˆ
*
x t
−
( )
ˆ
x t
≤
( )
q x t
1
ˆ
*
−
( )
ˆ
x t
+
1
∆ +
q c
2 2
q bM N
(23)
2
kt to t and taking into account of
∀ ≥
t
t
,
k
Integrating the above inequality from
(
*ˆ
x t
, the following inequality can be got:
(
ˆ
x t
=
)
)
k
k
( )
ˆ
*
x t
−
( )
ˆ
x t
≤
q
2
q
1
(
bM N
1
∆ +
c
2
(
)
e
(
q t
1
t
−
k
)
)
1
− =
δ
2
(
t
−
t
k
)
(24)
As a result, based on the triangle inequality and the inequalities (16) and (24),
it can be written that:
( )
e t
k
k
t
t
(
+
−
+
≤
+
≤
δ
2
( )
ˆ
x t
)
t
( )
ˆ
*
x t
)
t
,
−
( )
x t
−
(
(
e t
β
(25)
( )
ˆ
*
x t
−
)
(
δ
1
That finishes the proof of the theorem.
Theorem 1 indicates that, the upper bound of the estimated error depends on
, Lipschitz properties of
several factors including initial error of the states
the system and observer dynamics, sampling time of measurements ∆ and the
2c of magnitudes of disturbances
predictive horizon N, the bounds
and noise, as well as open-loop operation time of the observer
1c and
(
)ke t
t− .
−
)
t
t
t
k
k
k
Remark 3: Because the bound of
is the function of the observer's
open-loop operation time and the observer’s open-loop operation time is finite,
the function can be restricted to a region. We assume the region is
eΩ . It can be
derived that
( )
e t ∈Ω .
e
( )e t
ˆ
0
=
( )
u t
1
4. Lyapunov-Based Controller
)
(
( )
g x t
We assume that there exists a Lyapunov-based controller
1u for all ˆx inside a given stability re-
which satisfies the input constraints on
gion. And the origin of the nominal observer is asymptotically stable with
u = . From Lemma 1, this assumption indicates that there exist class
2
functions
and a continuous Lyapunov function V for the nominal ob-
server, which satisfy the following inequalities:
)
)
(
ˆ
V x
(
(
ˆ
F x g x
( )
iα ⋅
≤ −
α
2
α
3
,0,
)
)
(
)
(
≤
≤
)
)
ˆ
x
ˆ
x
y
ˆ
,
*
*
*
*
*
*
*
*
)
≤
α
4
(
*
ˆ
x
)
(26)
*
(
ˆ
x
α
1
(
ˆ
V x
∂
ˆ
*
x
∂
(
ˆ
V x
∂
ˆ
*
x
∂
(
ˆ
g x
*
)
∈
U
1
∀ ∈ ⊆
*ˆ
x D R
xn
for
the region
control law
where D is an open neighborhood of the origin. We denote
DρΩ ⊆ as the stability region of the nominal observer under the
u
1
u = .
2
and
(
ˆ
g x
)*
=
0
By continuity and the local Lipschitz property of F, it is obtained that there
exists a positive constant
2M such that:
(
( )
ˆ
F x t u t u t
( )
( )
,
,
1
2
,
(
y t
k
)
)
M≤
2
(27)
DOI: 10.4236/am.2017.812131
1839
Applied Mathematics
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