IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 7, JULY 2014

Sparse Packetized Predictive Control for Networked
Control Over Erasure Channels
Masaaki Nagahara, Member, IEEE,
Daniel E. Quevedo, Member, IEEE, andJan Østergaard, Senior Member, IEEE

Abstract—We study feedback control over erasure channels with packetdropouts. To achieve robustness with respect to packet-dropouts, the controller transmits data packets containing plant input predictions, which
minimize a finite horizon cost function. To reduce the data size of packets,
andwe propose to adopt sparsity-promoting optimizations, namely,-constrained
optimizations, for which efficient algorithms exist. We
show how to design the tuning parameters to ensure (practical) stabilityof the resulting feedback control systems when the number of consecutive
packet-dropouts is bounded.
Index Terms—Networked control systems (NCSs), packetized predictive
control (PPC).

I. INTRODUCTION
In networked control systems (NCSs) communication between
controller(s) and plant(s) is made through unreliable and rate-limited
communication links such as wireless networks and the Internet; see,e.g., [1]. Many interesting challenges arise and successful NCS design
methods need to consider both control and communication aspects.
In particular, so-called packetized predictive control (PPC) has beenshown to have favorable stability and performance properties, especially in the presence of packet-dropouts [2]–[4]. In PPC, the controller
output is obtained through minimizing a finite-horizon cost function
on-line and in a receding horizon manner. Each control packet contains a sequence of tentative plant inputs for a finite horizon of future
time instants and is transmitted through a communication channel.
Packets which are successfully received at the plant actuator side,
are stored in a buffer to be used whenever later packets are dropped.
When there are no packet-dropouts, PPC reduces to model predictive
control. For PPC to give desirable closed-loop properties, the more
unreliable the network is, the larger the horizon length (and thus thenumber of tentative plant input values contained in each packet) needs
to be chosen. Clearly, in principle, this would require increasing the
network bandwidth (i.e., its bit-rate), unless the transmitted signals
are suitably encoded. It is well-known that there exists a minimum
bit-rate for achieving stability of a networked feedback control system[1], [5]. The optimal quantizer for the minimum bit-rate is a dynamic
vector quantizer, and is, thus, hard to use in many applications. As
Manuscript received October 18, 2012; revised July 05, 2013; accepted
November 27, 2013. Date of publication December 11, 2013; date of current
version June 19, 2014. This work was supported in part under the JSPS
Grant-in-Aid for Scientific Research (C) 24560543, and also Australian
Research Council’s Discovery Projects funding scheme project DP0988601.
Recommended by Associate Editor G. N. Nair.
M. Nagahara is with the Graduate School of Informatics, Kyoto University,
Sakyo-ku Yoshida Honmachi, Kyoto, 606-8501, Japan (e-mail: nagahara@ieee.
org).
D. E. Quevedo is with the School of Electrical Engineering and Computer
Science, The University of Newcastle, Callaghan, Newcastle (NSW) 2308, Australia (e-mail: dquevedo@ieee.org).
J. Østergaard is with the Department of Electronic Systems, Multimedia
Information and Signal Processing (MISP), Aalborg University, Aalborg
DK-9220, Denmark (e-mail: jo@es.aau).
Color versions of one or more of the figures in this technical note are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2013.2294622

1899

an alternative, memoryless scalar quantizers, will often be preferable.
In this case, sparse representations [6] can be used to reduce thedata size of transmitted vectors in PPC. Sparse representations aim
at designing sparse vectors, which have few non-zero coefficients,
along with optimizing some performance indices. Since sparse vectors
contain many zero-valued elements, they can be easily compressedby only encoding a few nonzero coefficients and their locations witha memoryless scalar quantizer. Well-known examples of this kind of
encoding are JPEG in image processing [7] and algebraic CELP in
speech coding [8, Section 17.11.1]. Over the past few years, a numberof studies have been published which deal with sparsity for control,
including topics such as trajectory generation [9], state observation[10], [11], optimal control [12]–[18], and also sampled-data control[19]–[21].
The purpose of the present work is to introduce sparsity-promoting
optimizations for networked control with dropouts. We will show thatsparsity-promoting cost functions can be used in PPC to achieve good
control performance (as measured by a weighted quadratic norm of
the system state), whilst transmitting sequences with only few nonzero elements. By studying the sequence of optimal cost functions at
the instances of successful reception, we derive sufficient conditionsfor (practical) closed-loop stability in the presence of bounded packetdropouts.
It is well-known that sparsity-promoting optimization, which is often
described in terms of the so-called norm [6], is in principle hard tosolve due to its combinatorial nature [22]. However, there exist efficientmethods that compute the solution (or an approximation) of the optimization in the field of compressed sensing (see e.g., [23]). We will
focus on two such methods: One is convex relaxation where thenorm is used in place of the highly nonconvex norm. This leads tooptimization ( -regularized
optimization), which can be effectively solved with a fast algorithm called Fast Iterative ShrinkageThresholding Algorithm (FISTA) [24]. The other approach to obtainsparse solutions is through adoption of greedy algorithms. A greedy algorithm iteratively builds up the approximate solution of the -norm
optimization by updating the support set one by one. In particular, Orthogonal Matching Pursuit (OMP) [25] is quite simple and known tobe dramatically faster than exhaustive search.
Our present note complements our recent conference contribution
optimization for PPC. A limitation of the[26], which adopted an
approach in [26] is that for open-loop unstable systems, asymptotic stability cannot be obtained in the presence of bounded packet-dropouts;
the best one can hope for is practical stability. Our current technicalnote also complements the articles [27], [28], by presenting a detailed
technical analysis of the scheme, including proofs of results.
The remainder of this note is organized as follows: Section II revises basic elements of packetized predictive control. In Section III,
we show the motivation of sparsity-promoting optimization for PPC,
and formulate the design of the sparse control packets. In Section IV,
we study stability of the resultant networked control system. A numerical example is included in Section V. Section VI draws conclusions.
Notationfor. The identity matrix (of appropriate
We writedenotesdimensions) is denoted via . For a matrix (or a vector) ,and a positive
the transpose. For a vector
definite matrix, we define
,
,
, and
. The
of vector is defined as,
support set
and the “ norm” of is defined as
wheredenotes the cardinality of a set. Thus,is the number of nonzero

0018-9286 © 2013 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.
See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

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Fig. 1. Networked Control System with PPC. The dotted line indicates an erasure channel.

and allowed to depend on the state observation .
closed subset ofwill be given below.
More details on the choice of, we will investigate two
To obtain a sparse control vector
types of sparsity-promoting optimizations, namely, unconstrained
optimization and -constrained optimization.
Optimization

A. Unconstrained
elements in . For any Hermitian matrix ,denote the maximum and the minimum eigenvalues of.

and, respectively;

Here, the terminal and stage costs are given byand, where
,
, and
, and
is taken as(unconstrained). With the following notation:

II. PACKETIZED PREDICTIVE NETWORKED CONTROLLet us consider an unconstrained discrete-time linear time-invariant
plant model with a scalar input

..
.

(1)
where
,
. Throughout this work, we assume thatis reachable.
the pair
We are interested in an NCS architecture, where the controller
communicates with the plant actuator through an erasure channel,
as depicted in Fig. 1. This channel introduces packet-dropouts,
where
which we model via the dropout sequenceif packet-dropout occurs andotherwise. With
Packetized Predictive Control (PPC), as described, for instance,ofin [3], at each time instant , the controller uses the state
the plant (1) to calculate and sends a control packet of the form
to the plant input
node.
To achieve robustness against packet-dropouts, buffering is used.
, i.e.,
More precisely, suppose that at time instant , we haveis successfully received at the plant input side.
the data packet
Then, this packet is stored in a buffer, overwriting its previous contents.
is dropped, then the plant input
If the next packet
is set to, the second element of. The elements ofare then successively used until some packet,
is successfully received.
Remark 1: It is worth noting that (1) does not include disturbances.
Hence, as an alternative to PPC, one could simply transmit the system
state to the actuator and, upon successful reception, the actuator
could calculate and implement a semi-infinite plant input sequence.
In the present work, we focus on situations where the actuator does
not have sufficient computational capabilities precluding such an
open-loop control scheme. In contrast, the sparse PPC formulations
proposed in the present work provide feedback at all instances where
no dropouts occur. Our recent results concerning related schemes,see [3], [29], suggest that, in the presence of disturbances, PPC will
exhibit favorable robustness properties.
III. DESIGN OF SPARSE CONTROL PACKETS
In the present section we present two methods for the design ofinsparse PPC. The purpose is to obtain many zero elements, cf., [17], [18]. The control packet
the control packet
is designed at each time via a standard model predictive control formulation
(2)
where
and
are state and input predictions, respectively, defined
by
,
, and
.
The function defines the terminal cost and , the stage cost. In (2),, which is assumed to be a
we have introduced a constraint set

..
.

..

.

..
.

..
.

(3)

the optimization can be represented by(4)
optimization is known to produce a sparseThis (unconstrained)
vector via very efficient algorithms [23]. By simulation, we often obtainmuch sparser vectors than those produced by the method described
below (see also Section V). However, due to fundamental properties of
optimization, the state
never converges to the origin if
thethe plant (1) is unstable (see Proposition 4 below). Instead, in Theoremandin (3)
9 we will establish that, if design parametersconverges into aand (4) are appropriately chosen, then the stateclosed finite set including the origin.
B.

-Constrained

Here,
if
and
(2) is taken as

Optimization, whereas the stage cost is given by
if. The constraint set, used in(5)
are predicted states that depend on . We assume,, andare chosen such that the set
is. In terms of the notation (3), the optimization
non-empty for all
can, thus, be written as

where

(6)
The optimization (6) is in general extremely difficult to solve since
it requires a combinatorial search that explores all possible sparse sup. In fact, it has been proven to be NP hard [22], leading
ports ofto the development of sub-optimal algorithms; see, e.g., [23]. One approach to the combinatorial optimization is an iterative greedy algorithm called Orthogonal Matching Pursuit (OMP) [25].
Remark 2: In [17], [30]–[32], sparse control methods for closed loop
control without dropouts were studied, in essence, adopting an -constrained (or sparsity-constrained) optimization

where
is a positive integer less than . The optimization above
may be effectively solved via the CoSaMP algorithm described in [33].

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Since the bound ofis specified a priori, one can adopt the interleaved single pulse permutation (ISPP) design [8, Section 17.11.1]
for effectively encoding the support data of . A disadvantage of thisapproach is the difficulty in estimating a bound that guarantees stability of the feedback loop. In contrast, in the following section we willshow how design parameters in (2) can be chosen to ensure closed loop
stability in the presence of bounded dropouts.

where we used the norm inequality
for any, see [35]. This and the definition ofin (7) provide the upper. To obtain the lower bound, we simply note that by thebound on, we havefor any, anddefinition of.
hence
Remark 6: The least squares solution
approxiof thecost in (4) for given plant state
mates the optimizer. Since the following bound holds [36]:

IV. STABILITY ANALYSIS OF SPARSE PPC LOOPS
In this section, we provide stability results for sparse PPC with
optimization and -constrained optimization, as presented in Section III. To establish deterministic stability properties, we here impose
a bound on the maximum number of successive dropouts as follows:1
Assumption 3 (Bounded Packet-Dropouts): The number of consec.
utive packet-dropouts is uniformly bounded by
In view of the above, the horizon length in (2) allows one to tradecomputational complexity of the on-line optimization for robustness
with respect to dropouts. Thus, the less reliable the network is, theshould be chosen.
larger
A. Stability Analysis of

PPC

PPC, see (4). Our
We here analyze closed-loop stability ofanalysis uses elements of the technique introduced in [3]. A distinguishing aspect of the situation at hand is that, for open-loop unstable
plants, even when there are no packet-dropouts, asymptotic stability
will not be achieved, despite the fact that the plant-model in (1) is disturbance-free. In fact, we have the following proposition, which is directly proved from an equivalent dual problem of (4) (see [34, Appendix B]).
If
Proposition 4: Define, then.
It follows that if
and there are no dropouts at time , then. That is, the control system (1) behaves
the control will beas an open-loop system in the set . Hence, asymptotic stability willin general not be achieved, if has eigenvalues outside the unit circle.
This fundamental property is linked to sparsity of the control vector.
By the fact mentioned above, we will next turn our attention to practical stability (i.e., stability of a set) of the associated networked control
system. For that purpose, we will analyze the value function(7)

the upper bound forgiven in Lemma 5 will be tight, if

is small.

Having established the above preliminary results, we inwith the optimal vector
troduce the -th iterated mapping
defined in (4) through the recursion

This mapping describes the plant state evolution during periods of conis not a linear
secutive packet-dropouts. Note that, since the input
is nonlinear. The
function of (see Proposition 4), the function
following bound plays a crucial role to establish deterministic stability
guarantees:
satisfies theLemma 7 (Open-Loop Bound): Assume that
following Riccati equation:
(9)
,

with
we have. Then for any

and

,
(10)

and consider the sequence

Proof: Fix

where

(

and

) is given by, where. We then haveandis as in (4). First, we find bounds of.
where, we have
Lemma 5 (Bounds of ): For any

where, and the matrices

,
and

,
are given by

By the relation

andfor, we can bound the terms in the last sum above by(8)
Proof: Applying

to the cost in (7) gives

1Such a deterministic bound may arise for example in wired control networks

users, if the sole cause of dropouts is contention between users. Whenwitha priority-based, deterministic re-transmission protocol is used, transmission isintervals. Alternatively, in a wireless scenario withthen guaranteed within
random packet-dropouts, more than consecutive packet drops might trigger ahypothesis that there is a fault in the network, requiring a higher-level network
response outside the present formulation.

Thus, the cost function

can be upper bounded by(11). Sinceiswhere we have used the relation
among all ’s in, we havethe minimal value of, and hence inequality (10) holds. For the1902

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 59, NO. 7, JULY 2014case
, we consider the sequence.
, then (11) follows as in the case.
If we define
The above result can be used to derive the following contraction property of the optimal costs during periods of successive
packet-dropouts:. Assume thatsatisfies
Lemma 8 (Contractions): Let. Then there exists a real number(9) withsuch that for all, we havefrom Assumption 3, Lemma 8

controls. Since we have
gives

(16)
, and also forfor
we have,(17)

Now by induction from (17), it is easy to see that from Lemma 5
Proof: In this proof, we borrow a technique used in the proof of
we have[37, Theorem 4.2.5]. By Lemma 5, forThis inequality and (16) give the boundNow suppose that

. Thenand hence. From Lemma 7, it follows that:for, and this inequality also holds
for. Finally, by using the lower bound ofprovided in
Lemma 5, we have

with
(12)
Since
.
Next, consider the case where

,

, and

, it follows thatso that. This and Lemma 7 give

and

If

, then the above inequality also holds since.
We will next use Lemma 8 to establish sufficient conditions for pracPPC in the presence of packet-dropouts. Theorem
tical stability of
9 stated below shows how to design the parameters of the cost function
to ensure practical stability in the presence of bounded packet-dropouts
satisfying Assumption 3.
and choose
Theorem 9 (Practical Stability of - PPC): Letto satisfy (9) with. Thenis bounded, andfor all(13)
where is given in (12).
Proof: Denote the time instants where there are no packet, as
dropouts, i.e., where(14)
whereas the number of consecutive packet-dropouts is denoted via

where is defined in (13) and we used the inequality, for all. The above inequality leads to (13).
Theorem 9 establishes practical stability of the networked control
system. It shows that, provided the conditions are met, the plant state
will be ultimately bounded in a ball of radius . It is worth noting that,as in other stability results which use Lyapunov techniques, this boundwill, in general, not be tight.
B. Stability Analysis of

PPC

Here we analyze closed-loop stability of PPC, as described in (6).
Since stability will unavoidably be linked to feasibility, we begin withgiven in (5). Clearly, for given matrices
analysis of the feasible set
and , the feasible set
will be non-empty if the matrix
isgiven in (8), the “smallest”
which ensures that“larger” than
. In fact, we have:
Lemma 10: Let

For any, we have. Moreover, if,
is a closed, convex, and non-empty subset
then the feasible set.
of. Thenand hence
Proof: Suppose. This also implies thatis non-empty for any. Closedness and convexity ofare obvious since it).
is defined by a quadratic form (the set is a closed ellipsoid in
Based on this lemma, we hereafter assume that
(18)

(15)
Note that, with equality if and only if no dropouts occur
between instants
and
. Fix
and note that at time instant , the control packet is successfully transmitted to the buffer.
,consecutive
Then until the next packet is received at time
packet-dropouts occur. By the PPC strategy, the control input becomes,, and, since (1) is exact, theare determined by these open-loop
states

The feasible solutions for (6) can be characterized as follows:, there exists
Lemma 11 (Feasible Solutions): For any
such that(19)
where is given in (18).
Proof: The factgives the result.

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Remark 12: The error termin (19) may be interpreted as a“penalty charge” for sparsifying the vector (control packet) , sincewith the sparse control
will be larger
the term.
than with the least squares one,Now takearbitrarily and let
be the solution to the. Then, from Lemma 11 and wellRiccati equation (9) withknown results in dynamic programming [38, Chapter 3], all feasible
can be written as
control vectors

where
(20)
is the-th element ofsatisfying the inequality inand(19). The associated open-loop states are
(21)
where
(22)
By using the definition (20) of the matrix

Since
we finally obtain, we have. By mathematical induction,

(26)
.
where
Having established the contraction property (23), the following result shows that -PPC can be tuned to give asymptotic stability in thepresence of bounded packet-dropouts:
Theorem 14 (Asymptotic Stability): Suppose that the matrices , ,are chosen by the following procedure:
andarbitrarily.
1) Chooseto obtain.
2) Solve the Riccati equation (9) withandvia (24), (25), and (26).
3) Compute.
4) Choose such thatand set.
5) Compute,, optimizing (6) with
Then the sparse control packets
the above matrices, lead to asymptotic stability of the networked conas.
trol system, that is,
Proof: We use the notation of time instants where there are no. Since we havepacket-dropouts given in (14) and (15). Fix
from Assumption 3, Lemma 13 gives(27), we have. Also, for, the nextforinstant when the control packet is successfully transmitted, we have
whereis the-th row block of the matrix definedin (3).
as
For the -case, we use the quadratic functiona Lyapunov function candidate for the system sampled at the timesof successful transmission instants. The following result bears somesimilarities to Lemma 8, with the important difference that in (23) the
upper bound goes to zero as goes to the origin.
is chosen arbitrarily,
Lemma 13 (Contractions): Supposeis the solution of the Riccati equation (9) with, andis such that
. Let
. Then there exist
andsuch that
constants

It follows that at the time instants(no-dropout instants),
strictly decreases, and henceas. Then,(consecutive dropout instants),by (27), foris bounded by. Since the latter converges to zero,as.
we conclude that
In summary, the networked control system affected by boundedpacket-dropouts is asymptotically stable with the sparse control
are computed
packets obtained by the optimization (6) if , , andas per Theorem 14.

(23)
Proof: Substitution of the stateyields that

By the definition of

V. SIMULATION STUDY

given in (21) into

To assess the effectiveness of the proposed sparse control methods,
we consider a plant model of the form (1) with2in (22), we havewhere the last inequality is due to Lemma 11, and(24)
Then the above inequality gives(25)

We setand packet-dropouts are simulated with a model where
the probability distribution of the number of consecutive dropouts is. For the system above, we make simuniform overulation-based examination of the proposed stabilizing PPC formulaoptimization with
tions using sparsity-promoting optimizations,
optimization with OMP. We set the horizon length (orFISTA and.
the packet size) to
2The elements of these matrices are generated by random sampling from thenormal distribution with mean 0 and variance 1. Note that the matrix has 2) and 2 stable eigenvalues (0.4198
unstable eigenvalues (1.5259 and
).
and

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PPC, when the number of successive packet dropouts is bounded. Simulation results indicate that the proposed controllers provide, not only
stabilizing but also sparse control packets.
Future work may include obtaining analytical bounds on the sparsity of solutions. It is also of interest to apply the proposed control
methods to constrained nonlinear plant models with disturbances, andto channels with bit-rate limitations and unbounded packet-dropouts.
We foresee that this will require extending results in [4], [29] and also
the development of fast algorithms to solve the associated optimization
problems.
ACKNOWLEDGMENT
The authors wish to thank the Associate Editor and the anonymous
reviewers for valuable comments which have helped to improve thequality of this note.
Fig. 2. Averaged sparsity(bottom).
2-norm of the stateof the control vectors (top) and averaged

To compare these two sparsity-promoting methods with traditional
PPC approaches, we also consider the -optimal control

that minimizes, and the ideal least squares
solution, namely,
The regularization parameters for the
optimization and for
optimization are empirically chosen such that thenorm oftheand).
is minimized (the state
optimization, we choose the weighting matrix in (6) as
For the, and choose the matrix
according to the procedure shown in.
Theorem 14 with
With these parameters, we run 500 simulations with randomly generated packet-dropouts as described above, and with initial vector
in which each element is independently sampled from the normal distribution with mean 0 and variance 1. The top figure in Fig. 2 shows the
optimizaaveraged sparsity of the obtained control vectors. The, labeled by L1/L2 (i), always produces sparser
tion with
control vectors than OMP. Clearly, this property depends on how theis chosen. In fact, if we choose theregularization parameter, then the sparsity approximates that of OMP, assmaller value,
the curve labeled by L1/L2 (ii) in Fig. 2. On the other hand, if we use, then the control vector becomes . This isa sufficiently large
indeed the sparsest control, but leads to very poor control performance:
the state diverges until the control vector becomes nonzero (in accordance with the stability results established in Section IV). The bottom
as a funcfigure in Fig. 2 shows the averaged 2-norm of the statetion of for all 5 designs.
optimization based PPC,
We see that, with exception of thethe NCSs are all asymptotically stable. (Our simulation results even
optimization ofsuggest exponential stability.) In contrast, if the[26] is used, then only practical stability is observed. Note that thehas almost the same sparsity as OMP, but
optimization withthe response does not show asymptotic stability while OMP does. The
simulation results are consistent with Theorem 9 and Theorem 14.
VI. CONCLUSION
We have studied packetized predictive control formulations withand -constrained opsparsity-promoting cost functions (i.e.,
timizations) for networked control systems with packet dropouts. We-ophave established sufficient conditions for practical stability oftimal PPC and for asymptotic stability of -constrained -optimal

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