1
Optimal Integrated Control and Scheduling of
Networked Control Systems with Communication
Constraints: Application to a Car Suspension
System
Mohamed El Mongi Ben Gaid, Arben C¸ ela and Yskandar Hamam
Abstract— This paper addresses the problem of the optimal
control and scheduling of Networked Control Systems over
limited bandwidth deterministic networks. Multivariable linear
systems subject to communication constraints are modeled in
the Mixed Logical Dynamical (M LD) framework. The trans-
lation of the MLD model into th e Mixed Integer Quadratic
Programming (MIQP) formulation is described. This formulation
allows the solving of the optimal control and scheduling problem
using efficient branch and b ou nd algorithms. Advantages and
drawbacks of on-line and off-line scheduling algorithms are
discussed. Based on thi s discussion, a comput ation all y efficient
on-line scheduling algorithm, which can be seen as a compromise,
is presented and i ts performance is evaluated. Finally, t his
algorithm, called Optimal Pointer Placement (OPP) scheduling
algorithm, is applied to the control and scheduling of a car
suspension system.
Index Terms— Networked Control Systems, Limited Commu-
nication Control, Hybrid Systems, Control an d Scheduling Co-
design, Real-Time S cheduling, Active Suspen sion Control
I. INTRODUCTION
S
HARED communication networks are increasingly being
used to support the information exchange in distributed
control systems. Using a control network has many advan-
tages, such as a higher reliability, an easier deployment and
maintenance. However, many communication networks are
subject to bandwidth constraints (for example Underwater
Acoustic Networks [1] or Wireless Networks [2]). Reasons
behind these resource constraints are multiple. Guaranteeing
deterministic real-time communications induces important re-
strictions on the available bandwidth, especially when the
communication channel is noisy. In other situations, some
nodes of the communication network may be autonomous
and battery-powered. The lifetime of the batteries limits the
energy used in the transmission and thus limits the bandwidth
of the communications. On the other hand, in the automotive
industry, an increasing number of applications are being de-
veloped in order to improve the driving safety and comfort.
These applications need to share and to exchange an important
This work was supported by the COSI Laboratory, ESIEE Paris.
M-M. Ben Gaid and A. C¸ ela are with the COSI Laboratory, ESIEE Paris,
Cit´e Descartes, BP 99, 93162 Noisy-Le-Grand Cedex, FRANCE (e-mail:
{bengaidm,celaa}@esiee.fr)
Y. Hamam is with the A2SI Laboratory, ESIEE Paris, Cit´e Descartes, BP
99, 93162 Noisy-Le-Grand Cedex, FRANCE and with the LIRIS Laboratory,
UVSQ, 10/12 Avenue de l’Europe, 78140 Velizy, FRANCE (e-mail:
hamamy@esiee.fr)
amount of information. As a consequence, the local bus is
becoming more and more loaded. Using a more expensive
technology can solve these problems, but components price is
strongly balanced by production cost requirements.
Many approaches were proposed to address the problems
stemming from the use of communication networks in control
loops. Recently, it was shown that considering jointly control
and communication/computation resource allocation leads to
an improvement of the control performance with respect to the
classical approaches, given the same resources ( [3]–[11]).
The experimental study of communication networks char-
acteristics was performed in [12]. It was shown that the
transmission time of a message in the most used networks
can be neglected and that the delays occurring in networked
control loops are mainly due to the contention between the
different messages sent by the nodes of the network. The most
efficient way of reduction of these delays is through the design
of appropriate message scheduling strategies.
Scheduling algorithms for networked control applications
can be classified according to the scheduling policy, which
may be either off-line or on-line. In off-line scheduling al-
gorithms, the order of the different messages is specified at
design time. In on-line scheduling algorithms, the access to the
shared resource (the network) is determined at runtime, based
on the information related to the priorities of the messages
or to the controlled system state. Scheduling algorithms can
also be designed to tackle the scheduling of sensor measures
that are sent to the controller ( [6], [13]) or to address the
scheduling of control commands to the actuators ( [7]–[9]).
The problem of the optimal control and off-line scheduling
was studied in [7]. In the proposed model, the commands are
sent to the actuators through a shared TDMA bus. At each
slot, only one control command can be sent, the remaining
commands for the other actuators are held constant. The choice
of which actuator to update at each slot is handled using the
notion of communication sequence [14]. Only periodic com-
munication sequences were considered. Control commands
and periodic communication sequences are obtained through
the solving of a complex combinatorial optimization problem,
which aims at optimizing a quadratic cost function. This
problem can be treated using the approach developed in [15],
which has the advantage of applying the dynamic program-
ming method leading to a more efficient search heuristics.
On-line scheduling of control commands to the actuators
2
was studied in [8]. In the proposed model, it is assumed
that every slot, only one command vector can be sent to an
actuator group, the other control vectors are set to zero. The
stabilization is achieved using a Model Predictive Controller,
which calculates on-line the appropriate control law and the
allocation of the shared bus. The cost function used by the
MPC calculates a weighted sum of the infinity norms of the
states and the control commands over a specified horizon.
The optimization problem solved at each step by the MPC
algorithm was proven to be equivalent to the Generalized Lin-
ear Complementarity Problem [16]. However, setting control
commands to zero can degrade the dynamics of the controlled
continuous time systems, that’s why this approach is mainly
dedicated to discrete-time systems, rather than sampled-data
systems. The receding horizon approach was also addressed
in [9]. The control commands were assumed to belong to a
finite set of values. The solving of the optimization problem
involves the search of the Nearest Neighbor Vector Quantizer,
which may be very complex as the size of the finite set of
control values increases.
In this paper, the problem of the distributed control over de-
terministic real-time networks is addressed. The corresponding
optimal control and scheduling problem is formulated. An ef-
ficient approach for the solving of this problem is proposed. A
model predictive controller based on this problem formulation
is studied and evaluated. In opposition to [8], control signals
that could not be updated are held constant, a quadratic cost
function is used to evaluate the control performance and the
ability of the adaptive scheduling to improve the performance
of sampled-data systems is demonstrated. The on-line solving
of the optimization algorithm is very costly, that’s why a
computationally efficient on-line scheduling algorithm, which
can be seen as a compromise between off-line and on-line
scheduling algorithms, is designed and applied to a typical
example of a distributed control system: the active suspension
of car.
This paper is organized as follows. In section II, the mod-
eling of a multivariable control system with communication
constraints in the Mixed Logical Dynamical framework is
described. Section III addresses the translation of this model
into the Mixed Integer Quadratic Programming Formulation.
The finite-time optimal control and scheduling problem based
on this formulation is then solved. In section IV, a Model
Predictive Controller based on the formulated optimization
problem is designed. In section V, a computationally efficient
algorithm (the Optimal Pointer Placement (OPP) scheduling
algorithm) is proposed and compared to the Model Predictive
Controller. Finally, in section VI, the OPP algorithm is applied
to an active suspension system.
II. PROBLEM FORMULATION
Consider the continuous-time LTI plant described by:
˙x
c
(t) = A
c
x
c
(t) + B
c
u
c
(t) (1)
where x
c
(t) ∈ R
n
and u
c
(t) ∈ R
m
. The plant contains m
distinct actuators which are spatially distributed. The actua-
tors are connected to the main controller through a limited
bandwidth communication network (figure 1).
Plant
Limited bandwidth communication network
Controller
A
m
A
1
Fig. 1. A networked control system with communication constraints affecting
the transmission of control commands to the actuators
In order to derive a digital control law, a discrete-time
representation of system (1) at the sampling period T
s
is
considered:
x(k + 1) = Ax(k) + Bu(k) (2)
where x(k) = x
c
(kT
s
) and u(k) = u
c
(kT
s
).
In this article, it is assumed that the pair (A, B) is reachable
and that the full state vector x(k) is available to the controller
at each sampling period.
The communication resource is limited in the sense that,
at each sampling interval, it can carry at most b control
commands, where b ≤ m [4]. In contrast to [7], [9], where
only a control signal can be updated at each slot time (which
is equal to the sampling period), the adopted modeling allows
to specify separately the temporal parameters that are related
to the dynamics of the control system (the sampling period)
and those corresponding to the network capacity (the network
bandwidth), and thus permits to achieve a maximal use of the
available network resources.
The description of the constraints affecting the transmission
of the control commands to the actuators can be performed us-
ing the notion of scheduling function. The scheduling function
δ(k) at the sampling period k is defined by:
δ
i
(k) = 1 if u
i
(k) is updated at instant k
δ
i
(k) = 0 otherwise
Bandwidth limitations can be specified by the following
equality:
m
X
i=1
δ
i
(k) = b (3)
The digital-to-analog converters, which are located at the
actuators, use zero-order-holders to maintain the last received
control commands constant until new control values are re-
ceived. Consequently, if a control command is not updated at
the k
th
sampling period, then it is held constant. This assertion
can be modeled by the logic formula:
δ
i
(k) = 0 =⇒ u
i
(k) = u
i
(k − 1) (4)
Let v(k) ∈ R
b
be the vector of control commands that are
sent to the actuators through the network at the k
th
sampling
period and u
f
(k) the vector containing the b “free” elements of
u(k) (i.e. the elements of u(k) whose indices i satisfy δ
i
(k) =
1), arranged according to the increasing order of their indices.
3
The elements of v(k) are mapped to the b free elements of
u(k) such that:
u
f
(k) = v(k)
This mapping, together with constraint (4), can be described
by:
u
i
(k) = v
j
(k) if δ
i
(k) = 1 and
i
P
l=1
δ
l
(k) = j
u
i
(k) = u
i
(k − 1) otherwise
(5)
It can be easily verified that if v
j
1
and v
j
2
are mapped to
respectively u
i
1
and u
i
2
, than (j
1
< j
2
) implies that (i
1
< i
2
).
The mapping (5) can be written in matrix form. Let
[D
δ
(k)]
1≤i≤m,1≤j≤b
the matrix defined by:
[D
δ
(k)]
ij
= 1 if δ
i
(k) = 1 and
i
P
l=1
δ
l
(k) = j
[D
δ
(k)]
ij
= 0 otherwise
and:
E
δ
(k) = Diag (1 − δ
1
(k), . . . , 1 − δ
m
(k))
then:
u(k) = D
δ
(k)v(k) + E
δ
(k)u(k − 1)
Conversely, knowing the control input u(k) and the schedul-
ing decision δ(k), the vector of control commands v(k)
that were sent through the network can be determined. Let
[M
δ
(k)]
1≤i≤b,1≤j≤m
the matrix defined by:
[M
δ
(k)]
ij
= 1 if δ
j
(k) = 1 and
j
P
l=1
δ
l
(k) = i
[M
δ
(k)]
ij
= 0 otherwise
then:
v(k) = M
δ
(k)u(k) (6)
The control system and its communication network can
be seen as a hybrid system S, having two types of inputs:
continuous inputs v(k) (control commands transmitted through
the network) and logical inputs δ(k) (scheduling decisions)
(figure 2). The considered model of system S is composed of:
• A recurrent equation (2) describing the dynamics of the
system.
• An equality constraint (3) expressing the limitations of
the communication medium.
• Logic rules (5) modeling the zero-order holders and
addressing the mapping of control signals v
i
that are sent
through the network to control system inputs u
i
according
to the scheduling decisions δ
i
.
This model can be handled using the Mixed Logical Dynam-
ical (MLD) framework [17].
Network
δ(k)
u(k) x(k)
x(k + 1) = Ax(k) + Bu(k)
v(k)
Fig. 2. Hybrid model of the networked control system S
III. FINITE-TIME OPTIMAL CONTROL AND SCHEDULING
In this paragraph, the problem of the finite-time optimal
control and scheduling is formulated and translated into the
Mixed Integer Quadratic Programming (MIQP) formulation.
It is assumed that u(k) = 0 and v(k) = 0 for k < 0, and
that control commands u(k) and v(k) are subject to saturation
constraints:
L
i
≤ u
i
(k) ≤ U
i
and L
i
≤ v
i
(k) ≤ U
i
where L
i
< 0 and U
i
> 0.
A. Performance index definition
In order to quantify the “Quality” of the control and schedul-
ing, a quadratic cost function is associated to system (1) :
J
c
(x
c
, u
c
, 0, T
f
) =
T
f
Z
0
x
T
c
(t)Q
c
x
c
(t) + u
T
c
(t)R
c
u
c
(t)
dt
+ x
T
c
(T
f
)S
c
x
c
(T
f
)
(7)
where T
f
= NT
s
and Q
c
, R
c
and S
c
are positive definite
matrices. These matrices define the design specifications of
the ideal controller. The sampled-data representation of the
cost function J
c
(x
c
, u
c
, 0, T
f
) at the sampling period T
s
is:
J(x, u, 0, N) =
N−1
X
k=0
x(k)
u(k)
T
Q
1
Q
12
Q
T
12
Q
2
x(k)
u(k)
+ x
T
(N)Q
0
x(N)
(8)
The expressions of Q
1
, Q
2
, Q
12
and Q
0
can be found in ( [18],
pp. 411–412). Note that this representation does not involve
any approximation and is exact. In the following, it is assumed
that Q, Q
2
and Q
0
are positive definite matrices, where:
Q =
Q
1
Q
12
Q
T
12
Q
2
B. Formalization and solving of the finite-time optimal inte-
grated control and scheduling problem
The finite-time optimal control and scheduling problem can
be formalized as follows:
Problem formulation 1: Given an initial state x(0) and a
final time N, find the optimal control sequence v
N−1
=
(v(0), . . . , v(N − 1)) and the optimal scheduling sequence
δ
N−1
= (δ(0), . . . , δ(N − 1)) which minimize the perfor-
mance index:
J(x, u, 0, N) =
N−1
X
k=0
x(k)
u(k)
T
Q
x(k)
u(k)
+x
T
(N)Q
0
x(N)
subject to:
x(k + 1) = Ax(k) + Bu(k)
m
X
i=1
δ
i
(k) = b
u(k) = D
δ
(k)v(k) + E
δ
(k)u(k − 1)
L
i
≤ v
i
(k) ≤ U
i
4
System S is time varying and the problem of finding the
optimal control sequence v
N−1
for a given fixed scheduling
sequence δ
N−1
is a quadratic programming (QP) problem.
The number of possible scheduling sequences is finite. The
resolution of problem 1 is reduced to the exploration of all the
feasible scheduling sequences and the solving of a QP problem
for each fixed scheduling sequence. However, in practice, the
number of feasible scheduling sequences grows exponentially
with N , which means that exhaustive search cannot be applied
to problems with large values of N.
The solution of problem 1 can be obtained through the
solving of a simpler optimization problem, which can be seen
as a constrained control problem, where the variables v
N−1
are eliminated and the constraint (5) is replaced by (4). Let
u
N−1
= (u(0), · · · u(N − 1)), this problem can be stated as
follows:
Problem formulation 2: Given an initial state x(0) and a
final time N , find the optimal control sequence u
N−1
and
the optimal scheduling sequence δ
N−1
which minimize the
performance index:
J(x, u, 0, N) =
N−1
X
k=0
x(k)
u(k)
T
Q
x(k)
u(k)
+x
T
(N)Q
0
x(N)
subject to:
x(k + 1) = Ax(k) + Bu(k)
m
X
i=1
δ
i
(k) = b
δ
i
(k) = 0 =⇒ u
i
(k) = u
i
(k − 1)
L
i
≤ v
i
(k) ≤ U
i
In order to solve this problem, it is necessary to translate
the logical formula (4) into linear inequalities. The connective
“=⇒” can be eliminated if (4) is rewritten in the equivalent
form:
u
i
(k) − u
i
(k − 1) = δ
i
(k)u
i
(k) − δ
i
(k)u
i
(k − 1) (10)
However, equation (10) contains terms which are the product
of logical variables and continuous variables. The use of the
procedure described in [17] allows the translation of this
product into an equivalent conjunction of linear inequalities.
For example, let:
z
i
(k) = δ
i
(k)u
i
(k) (11)
Then (10) can be rewritten in the equivalent form:
z
i
(k) ≤ U
i
δ
i
(k)
z
i
(k) ≥ L
i
δ
i
(k)
z
i
(k) ≤ u
i
(k) − L
i
(1 − δ
i
(k))
z
i
(k) ≥ u
i
(k) − U
i
(1 − δ
i
(k))
(12)
Note that the same procedure can be applied to w
i
(k) =
δ
i
(k)u
i
(k − 1).
Let ∆ =
δ(0)
.
.
.
δ(N − 1)
, U =
u(0)
.
.
.
u(N − 1)
, X =
x(0)
.
.
.
x(N)
,
Z =
z(0)
.
.
.
z(N − 1)
, W =
w(0)
.
.
.
w(N − 1)
and V =
∆
U
X
Z
W
,
then problem 2 can be written:
(
min
V
1
2
V
T
HV + f
T
V
AV ≤ B
(13)
where H, f, A, and B can be easily deduced form the
discussion above. Problem (13) is a Mixed-Integer Quadratic
Program. The advantage of this formulation is the existence
of many efficient academic and commercial solvers, based on
the branch and bound algorithm.
Problem 1 is identical to problem 2 augmented with the
additional constraint: v(k) = u
f
(k). As a consequence, the
optimal solutions of problem 1 can be deduced from the
optimal solutions of problem 2 using the mapping (6).
IV. MODEL PREDICTIVE CONTROL
Open-loop optimization problems constitute the cornerstone
of a successful control method: the model predictive control
(MPC). MPC has strong theoretical foundations, and many
interesting properties which make it suitable to address con-
strained control problems. However, its main drawback is that
it requires very expensive computing resources, which make
it only applicable to slow systems, like chemical processes.
Model predictive control is the standard approach to control
MLD systems. Its application to this particular problem was
motivated by:
• The need to optimize simultaneously control actions and
network scheduling, in order to achieve a better quality
of control than the static network allocation schemes.
• The need for a control law that changes on-line the
sampling period in order to improve the quality of control.
This requires that these variations are taken into account
by the control law [19].
Using Model Predictive Control, an optimal control problem
is solved on-line at each sampling period. It aims at finding
the optimal control values sequence ˆu
N−1
= (ˆu(0), ..., ˆu(N −
1)) and the optimal network allocation sequence
ˆ
δ
N−1
=
(
ˆ
δ(0), ...,
ˆ
δ(N − 1)) which are solutions of the following
optimization problem:
min
ˆu
N −1
,
ˆ
δ
N −1
N−1
P
h=0
ˆx(h)
ˆu(h)
T
Q
ˆx(h)
ˆu(h)
+ ˆx
T
(N)Q
0
ˆx(N )
subject to:
ˆx(0) = x(k)
ˆx(h + 1) = Aˆx(h) + Bˆu(h) , h ∈ [0, N − 1]
m
P
i=1
ˆ
δ
i
(h) = b , h ∈ [0, N − 1]
ˆ
δ
i
(0) = 0 =⇒ ˆu
i
(0) = u
i
(k − 1)
ˆ
δ
i
(h) = 0 =⇒ ˆu
i
(h) = ˆu
i
(h − 1) , h ∈ [1, N − 1]
(14)
The solution of this problem is based on the prediction of the
future evolution of the system over a horizon of N sampling
periods. This predicted evolution is calculated according to
the model of the plant, knowing the current state x(k) of the
5
system and the previously applied control input u(k − 1). The
variables ˆx(h), h ∈ [0, N] represent the predicted values of
system states x(k + h). The sequences (ˆu(0), ..., ˆu(N − 1))
(virtual control sequence) and (
ˆ
δ(0), ...,
ˆ
δ(N − 1)) (virtual
network allocation sequence) are called virtual sequences,
because they are based on the predicted evolution of the
system. The resolution of this problem aims at finding the
optimal virtual control sequence (ˆu
∗
(0), ..., ˆu
∗
(N −1)) and the
optimal virtual network allocation (
ˆ
δ
∗
(0), ...,
ˆ
δ
∗
(N −1 )) which
minimize a quadratic cost function over a finite horizon of N
sampling periods. Assuming that the optimal virtual sequences
exist, the actual control commands are obtained by setting:
v(k) = M
ˆ
δ
∗
(0)ˆu
∗
(0) (15)
and:
δ(k) =
ˆ
δ
∗
(0) (16)
and disregarding the remaining elements of the sequences
(ˆu
∗
(1), ..., ˆu
∗
(N − 1)) and (
ˆ
δ
∗
(1), ...,
ˆ
δ
∗
(N − 1)). At the next
sampling period (step k+1), the whole optimization procedure
is repeated, based on x(k + 1).
A important issue concerns the stability of the proposed
Model Predictive Controller. If the following constraint is
added to problem (14):
ˆx(N ) = 0 (17)
the following result is obtained:
Theorem 1: If at k = 0, a feasible solution exists for the
problem (14) augmented with the additional constraint (17),
then ∀Q = Q
T
> 0, the MPC law (14)(17) stabilizes the
system S such that:
lim
k→∞
x(k) = x
e
= 0 and lim
k→∞
u(k) = u
e
= 0
Proof: The proof can be easily performed following the
same ideas of the proof of the sufficient stability conditions for
the Model Predictive Control of MLD systems stated in [17].
Although the application of this technique gives very good
results (as it will be illustrated in the next section), its major
drawback is that it requires very expensive computational
resources, which makes its application to fast systems imprac-
ticable. A more efficient heuristics are needed.
V. OPTIMAL POINTER PLACEMENT SCHEDULING
The motivation behind the Optimal Pointer Placement
(OPP) scheduling algorithm presented in this section is to be a
compromise between the advantages of the on-line scheduling
(control performance) and those of the off-line scheduling (a
very limited usage of computing resources).
A. Algorithm description
First, a time varying state representation of system S is
derived. Let ξ(k) = u(k − 1) and ˜x(k) =
x(k)
ξ(k)
. For a fixed
scheduling function δ verifying the constraint (3), the system
S can be represented by the time-varying state equation:
˜x(k + 1) =
˜
A(k)˜x(k) +
˜
B(k)v(k) (18)
where:
˜
A(k) =
A BE
δ
(k)
0 E
δ
(k)
and
˜
B(k) =
BD
δ
(k)
D
δ
(k)
Assume that a periodic off-line controller as well as a
periodic off-line schedule (both of period T ) guaranteeing the
asymptotic stability of the system exist. The periodic controller
is defined by a periodic sequence of state feedback control
gains K
T −1
= (K(0), . . . , K(T −1)) and the schedule by the
periodic communication sequence γ
T −1
= (γ(0), . . . , γ(T −
1)). Note that optimal K
T −1
and γ
T −1
selection is described
in [7] (considering worst case initial conditions) and in [15]
(according to LQG arguments).
At runtime, the execution of the periodic off-line controller
and scheduler can be described using the notion of pointer. The
pointer can be seen as a variable which contains the index of
the control gain to use and the scheduling to apply. The pointer
is incremented at each sampling period. If it reaches the end
of the sequence, its position is reset. More formally, if the
pointer is started at position p (0 ≤ p < T ) its expression
I
p
(k) at the k
th
sampling period is:
I
p
(k) = (k + p) mod T (19)
According to the off-line strategy, the control commands that
are sent and the scheduling decisions are chosen such that:
v(k) = K(I
p
(k))˜x(k) and δ(k) = γ(I
p
(k))
The idea behind the OPP scheduling heuristic is that instead
of finding an optimal solution to problem (14), the search is
restricted to the finding of a sub-optimal solution, based on an
optimal off-line schedule, over a horizon N (which is assumed
to be a multiple of T ), according to the following problem:
min
p
N−1
P
h=0
ˆx(h)
ˆu(h)
T
Q
ˆx(h)
ˆu(h)
+ ˆx
T
(N)Q
0
ˆx(N)
subject to:
ˆx(0) = x(k)
ˆu(−1) = u(k − 1)
and for all h ∈ [0, N − 1]
ˆv(h) = K(I
p
(h))
ˆx(h)
ˆu(h − 1)
ˆu(h) = D
γ
(I
p
(h))ˆv(h) + E
γ
(I
p
(h))ˆu(h − 1)
ˆx(h + 1) = Aˆx(h) + Bˆu(h)
(20)
The cost function is calculated according to a prediction of
the future evolution of the system (described by ˆx(h)). This
evolution is calculated assuming that sequences K
T −1
and
γ
T −1
are started from position p. The solution of problem (20)
is the pointer’s position p
∗
which minimizes the cost function
ˆ
J(˜x(k), p), subject to the constraints expressed above, where:
ˆ
J(˜x(k), p) =
N−1
X
h=0
ˆx(h)
ˆu(h)
T
Q
ˆx(h)
ˆu(h)
+ ˆx
T
(N)Q
0
ˆx(N)
The control command v(k) = K(p
∗
)˜x(k) is sent according to
the scheduling function δ(k) = γ(p
∗
).
