Research A rticle
Improved Distributed Model Predictive Control with
Control Planning Set
Wei Chen
Department of Automation, Hefei University of Technology, Hefei 230009, China
Correspondence should be addressed to Wei Chen; windysunny@gmail.com
Received June ; Revised August ; Accepted August
Academic Editor: Yuanyuan Zou
Copyright © Wei Chen. is is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We focus on distributed model predictive control algorithm. Each distributed model predictive controller communicates with the
others in order to compute the control sequence. But there are not enough communication resources to exchange information
between the subsystems because of the limited communication network. is paper presents an improved distributed model
predictive control scheme with control planning set. Control planning set algorithm approximates the future control sequences
by designed planning set, which can reduce the exchange information among the controllers and can also decrease the distributed
MPC controller calculation demand without degrading the whole system performance much. e stability and system performance
analysis for distributed model predictive control are given. Simulations of the four-tank control problem and multirobot multitarget
tracking problem are illustrated to verify the eectiveness of the proposed control algorithm.
1. Introduction
Model predictive control (MPC), also referred to receding
horizon control (RHC), is an attractive control strategy
because of its ability to control systems with input and output
constraints in the optimization problem. e input sequence
is calcul ated by solving an optimization problem (minimiza-
tion of a given performance index) over a prediction horizon.
Once the optimization problem is solved, only the rst input
value is implemented into the system. In the next sampling
time, a new optimization problem is solved repeatedly. MPC
has been widely applied in various control areas over the past
few decades [–].
Nowadays, systems are becoming more and more com-
plex. In centralized MPC, all the inputs sequences are
optimized with respect to one given performance index in
a single optimization problem. However, when the number
of the state variables and inputs of the system becomes
larger and larger, the computation burden of the centralized
optimization problem may increase signicantly. Moreover,
theentiresystemwouldbeoutofcontrolifthecentralized
MPC controller fails. erefore it is impractical to apply
the centralized MPC to large-scale systems. In fact, a large-
scale system is composed by physically parted subsystems.
Many decentralized and distributed model predictive con-
trol (DMPC) algorithms have been recently proposed [–
],whicharesomefeasiblealternativestoovercomethe
computational burden of the centralized MPC.
In DMPC architecture, subsystems communicate with
eachothervianetworksandtheinputsarecomputedbysolv-
ing more than one optimization problem in each subsystem
in a coordinated fashion. ere are many achievements on
DMPC strategy and a survey of major DMPC algorithms is
presented in [, ]. e existing DMPC algorithms can be
divided into dierent categories.
Based on the topology of the communication network,
DMPC can b e divided into fu l ly connected algorithms and
partially connected algorithms. In fully connected algo-
rithms, DMPC is able to communicate with the rest of the
local controllers [, ]. In p art ially connected algorithms,
local optimization problems are s olved by taking into account
the neighboring (not the whole system) interaction and
Hindawi Publishing Corporation
Journal of Control Science and Engineering
Volume 2016, Article ID 8167931, 14 pages
http://dx.doi.org/10.1155/2016/8167931
Jo urnal of Control Science and Engineering
solution, which is suitable for loosely connected subsystems
[, ]. However, it will deteriorate the whole system perfor-
mance.
Basedontheexchangetimesamongthedistributedcon-
trollers, DMPC can be divided into noniterative algorithms
and iterative algorithms. In iterative algorithms, information
is transmitted among the DMPC controllers many times in
the sampling interval [, ]. On the contrary, in noniterative
algorithms DMPC controller communicates with the other
controllers only once in the sampling interval [, ].
In this article, we consider that the DMPC controllers
can exchange information only once while they are solving
their local optimization problems at each sampling time
and the connectivity of the communication is sucient
for the distributed controller to obtain information. is
paper proposes an extension of the fully connected nonit-
erative DMPC algorithm. However, the exchange informa-
tion between subsystems is usually realized over a digital
communication network. us, the local systems can only
have limited communication resource. F or example, in a
networked environment, bandwidth limitations can restrict
the amount of exchange information. us, it is necessary to
restrict the distributed controllers to exchange information.
e proposed DMPC in the paper reduces the communica-
tion information compared to the standard distributed MPC
control scheme in complex large-scale systems and at the
same time decreases computational burden of each controller.
is algorithm also provides a reasonable trade-o between
system performance and low communication requirements
needed to reach a cooperative solution.
e rest of the paper is organized as follows. In Section ,
the centralized and distributed model predictive control
problem is formulated. In Sec tion , the improved distributed
model predictive control with control planning set (CP-
DMPC) is proposed. e stability and performance analysis
is provided in Section . In S ection , the simulations of
theproposedcontrollertofour-tanksystemandmultirobot
multitarget tracking system are presented. Finally, the conclu-
sions of the work are given in Section .
2. Centralized and Distributed Model
Predictive Control Formulation
Without loss of generality, suppose that the whole system is
comprised of interconnected subsystems. And consider
that each subsystem only couples through the input [].
e discrete-time state-space model for th subsystem is as
follows:
𝑚,𝑖
(
+1
)
=
𝑚,𝑖
𝑚,𝑖
(
)
+
𝑚,𝑖𝑖
𝑖
(
)
+
𝑁
𝑗=1,𝑗 =𝑖
𝑚,𝑖𝑗
𝑗
(
)
,
(a)
𝑖
(
)
=
𝑚,𝑖
𝑚,𝑖
(
)
,
(b)
where = 1,...,.
𝑚,𝑖
(),
𝑖
(),and
𝑖
() are the state
vector, the control input vector, and the output vector of
th subsystem at kth sampling time. e model (a), (b)
is changed to suit the model predictive control design with
an embedded integrator. e augmented model of the th
subsystem state space model is
𝑖
(
+1
)
=
𝑖
𝑖
(
)
+
𝑖𝑖
𝑖
(
)
+
𝑖
(
)
,
(a)
𝑖
(
)
=
𝑖
𝑖
(
)
,
(b)
where a new state variable vector is chosen to be
𝑖
(
)
=
𝑚,𝑖
(
)
𝑖
(
)
()
and a new control variable vector is chosen to be
𝑖
(
)
=
𝑖
(
)
−
𝑖
(
−1
)
()
and the dierence of the state variable is denoted by
𝑚,𝑖
(
+1
)
=
𝑚,𝑖
(
+1
)
−
𝑚,𝑖
(
)
.
()
e state interaction vector is given by
𝑖
(
)
=
𝑁
𝑗=1,𝑗 =𝑖
𝑖𝑗
𝑗
(
)
.
()
e triplet
𝑖
,[
𝑖𝑖
,
𝑖𝑗
],
𝑖
is
𝑖
=
𝑚,𝑖
𝑚,𝑖
𝑚,𝑖
,
𝑖𝑖
=
𝑚,𝑖𝑖
𝑚,𝑖
𝑚,𝑖𝑖
,
𝑖𝑗
=
𝑚,𝑖𝑗
𝑚,𝑖
𝑚,𝑖𝑗
,
𝑖
=
.
()
e model of the whole system (centralized model) can be
expressed in compact way
(
+1
)
=
(
)
+
(
)
,
(a)
(
)
=
(
)
(b)
Jo urnal of Control Science and Engineering
u(k)
y(k)
Index performance
of system Σ
Constraints of
system Σ
Optimization
CMPC
Subsystem i
Subsystem N
Inherent
interaction
Network
System Σ
Setpoint Σ
+
−
Model of
system Σ
y
1
(k)
y
i
(k)
y
N
(k)
u
1
(k)
u
i
(k)
u
N
(k)
···
Subsystem 1
u(k |k),...,u(k+N
p
−1|k)
y(k|k),...,y(k+N
p
−1|k)
F : Centralized MPC control system architecture.
with state vector () ∈
𝑛
𝑥
,controlinputvector() ∈
𝑛
𝑢
,andoutputvector() ∈
𝑛
𝑦
. A, B, and are the whole
system matrices. is implies that
=
1
d
𝑖
d
𝑁
,
=
11
...
1𝑗
...
1𝑁
.
.
. d
.
.
. d
.
.
.
𝑖1
...
𝑖𝑖
...
𝑖𝑁
.
.
. d
.
.
. d
.
.
.
𝑁1
...
𝑁𝑗
...
𝑁𝑁
,
=
1
d
𝑖
d
𝑁
,
(
)
=
1
(
)
,
2
(
)
,...,
𝑁
(
)
𝑇
,
(
)
=
1
(
)
,
2
(
)
,...,
𝑁
(
)
𝑇
,
(
)
=
1
(
)
,
2
(
)
,...,
𝑁
(
)
𝑇
.
()
2.1. Centralized Model Predictive Control Formulation. e
main idea of the centralized model predictive control for-
mulation is one large-scale optimization with constraint.
e centralized MPC control system architecture diagram is
showninFigure.
In the centralized model predictive control formulation,
at each sampling time centralized MPC controller obtains the
whole system measurement () =[
1
(),
2
(),...,
𝑁
()]
and the control objective minimizes the following global
performance index:
(
)
=
𝑁
𝑖=1
𝑖
(
)
,
𝑖
(
)
=
𝑁
𝑝
𝑙=1
𝑖
(
+|
)
−
𝑑
𝑖
(
+
)
2
𝑄
𝑖
+
𝑁
𝑢
𝑙=1
𝑖
(
+−1|
)
2
𝑅
𝑖
(a)
s.t.
𝑖
(
++1|
)
=
𝑖
𝑖
(
+|
)
+
𝑖𝑖
𝑖
(
+|
)
+
𝑖
(
+|
)
,
𝑖
(
+|
)
=
𝑖
𝑖
(
+|
)
=1,...,.
(b)
Her e
𝑝
is the prediction horizons and
𝑢
is t he control
horizons. And
𝑝
≥
𝑢
.
𝑖
and
𝑖
are penalties on the output
variables and control variables, respectively.
𝑑
𝑖
is the output
set point. And because the central controller can handle all
th e i nform ati on of th e s ystem , t he inte rac tion pre dic tions
𝑖
(+|)are known at time .
is optimization problem (a), (b) can be solved by a
standard quadratic program algorithm with constraints. e
optimal control sequence
∗
(,
𝑢
|)=[
∗
( | ),
Jo urnal of Control Science and Engineering
Subsystem 1
Index performance
of subsystem 1
Constraints of
subsystem 1
Optimization 1
Model of
subsystem 1
Setpoint 1
Subsystem N
Index performance
of subsystem N
Constraints of
subsystem N
Optimization
N
Model of
subsystem
N
Setpoint N
Subsystem i
Index performance
of subsystem i
Constraints of
subsystem i
Optimization i
Model of
subsystem i
Setpoint i
DMPC-1
DMPC-i
DMPC-N
Controller information
exchange
Controller information
exchange
Inherent
interaction
Inherent
interaction
···
···
···
···
+
−
u
1
(k) y
1
(k)
y
i
(k)
y
N
(k)
+
−
u
i
(k)
+
−
u
N
(k)
y
1
(k)···y
N
(k)
y
1
(k)···y
N
(k)
u
1
(k | k), . . . , u
1
(k + N
p
−1|k)
u
1
(k | k), . . . , u
1
(k + N
p
−1|k)
y
1
(k|k),...,y
1
(k + N
p
−1|k)
u
N
(k | k), . . . , u
N
(k + N
p
−1|k)
u
i
(k|k),...,u
i
(k + N
p
−1|k)
y
i
(k | k), . . . , y
i
(k + N
p
−1|k)
u
1
(k|k),...,u
1
(k + N
p
−1|k)
u
N
(k|k),...,u
N
(k + N
p
−1|k)
u
N
(k | k), . . . , u
N
(k + N
p
−1|k)
y
N
(k|k),...,y
N
(k + N
p
−1|k)
F : DMPC control system architecture.
∗
(+1|),...,
∗
( +
𝑢
−1|)]is calculated and
only the rst control signal
∗
( | )=[
∗
1
( | ),
∗
2
( |
),...,
∗
𝑁
(|)] isappliedtothewholesystem;aernew
measurements are available, a new optimization problem is
solved in the next sampling time.
Many engineering applications such as power systems,
unmanned aerial vehicles, sensor networks, economic sys-
tem, transportation systems, and process control systems,
have become larger and more complex. e overall number
of inputs and states (outputs) is very large, and the optimized
control sequence
∗
(,
𝑢
|)is highly dimensional.
A single optimization problem may require computational
resources (CPU time, memory, etc.). In view of the above
consideration, it is natural to look for distributed MPC
algorithms.
2.2. Distributed Model Predictive Control Formulation. In the
distributed model predictive control formulation, the large
size optimization problem is replaced by small ones that
work cooperatively towards achieving the performance of
centralized control system. And the following assumptions
are made.
(a) Predictive horizons
𝑝
and control horizons
𝑢
are
thesameforeachsubsystem.
(b) Controllers are synchronous.
(c) Controllers communicate with each other only once
within a sampling time inter val.
(d) Controllers are interconnected and can obtain infor-
mation which the controllers need.
And the DMPC control system architecture diagram is
showninFigure.
Jo urnal of Control Science and Engineering
➀ Sensor measurement delay
➂ Controller information communication delay
k
k+1
➀➁ ➂
➁ DMPC-i (i=1,...,N)controller calculation delay
F : Delay time analysis per sampling interval.
e th subsystem minimizes t he following local perfor-
mance index, which is the th optimization problem []:
𝑖
(
)
=
𝑁
𝑝
𝑙=1
𝑖
(
+|
)
−
𝑑
𝑖
(
+
)
2
𝑄
𝑖
+
𝑁
𝑢
𝑙=1
𝑖
(
+−1|
)
2
𝑅
𝑖
(a)
s.t.
𝑖
(
++1
)
=
𝑖
𝑖
(
+
)
+
𝑖𝑖
𝑖
(
+
)
+
𝑖
(
+−1
)
,
𝑖
(
+
)
=
𝑖
𝑖
(
+
)
.
(b)
Itcanbeseenthattheglobalperformanceindexcanbe
decomposed into a number of local performance indexes, but
theoutputofeachagentisstillrelatedtoalltheinputvariables
due to the input coupling. Because controllers communicate
with each other only once within a sampling time interval,
the interaction predictions
𝑖
(+|)are unknown for
the th subsystem. And only the prediction
𝑖
(+|−
1) based on the information broadcasted at time −1is
available. A noniterative algorithm is developed to seek the
distributed solution at each sampling time. Based on the
information from other subsystems, each controller solves
local optimization problems to determine the future sequence
∗
𝑖
(,
𝑢
|)=[
∗
𝑖
( | ),
∗
𝑖
(+1|),...,
∗
𝑖
(+
𝑢
−1|)]and broadcast
∗
𝑖
( | )by communication
network to the other controllers.
3. Improved Distributed Model
Predictive Control with Control Planning
Set (CP-DMPC)
Besides the computational advantages of DMPC, the amount
of data needs to be exchanged among distributed controllers.
In the paper, fully connected noniterative DMPC algorithm
is focused on. However, each system exchanges information
with each other by both their initial state and their optimized
input. And time delays exist in communication network. In
Figure , we can see that time delay consists of three parts,
1 2 3 4 5 6 7 8
0
2
4
6
8
10
12
Control horizon
Control value
MPC
CP MPC, 𝛽=1
CP MPC, 𝛽 = 1.1
CP MPC, 𝛽 = 0.9
F : e comparison between traditional MPC and CP MPC.
sensor measurement delay, DMPC controller calculation
delay, and controller information communication delay.
In this paper, a control planning set algorithm is com-
bined with DMPC controller to reduce the controller infor-
mation communication delay and meanwhile it also can
decrease the DMPC controller calculation demand without
degrading the whole system performance much. e control
planning set method presented in the paper is inspired by the
pulse-step control strategy []. Suboptimal strategies can be
obtained by restricting the future control sequence
(
+|
)
=
(
(
+−1|
))
.
()
For specication and simplicity, we choose function as
a linear function:
(
+|
)
=
(
+−1|
)
.
()
In the control planning set algorithm, the future control
sequence is restricted by one possibility. e parameter
is chosen to plan the future control sequence increases or
decreases in the same direction, which is suitable for the
experience of control engineering. And it will prevent the
frequent oscillation of the control input; see Figure .
In a traditional MPC scheme, the optimized control
sequence is calculated via the performance index, which may
oscillate during the control horizon. In CP
MPC scheme,
the optimized control sequence changes in one direction,
which may not obtain the optimum solution but is suitable
for the control engineering. In control engineering, in some
time period control value does not change suddenly and
frequently, and this is good for the control hardware device.