Packetized Predictive Control for Rate-Limited Networks via
Sparse Representation
Masaaki Nagahara Member, IEEE, Daniel E. Quevedo, Member, IEEE,
and Jan Østergaard, Senior Member, IEEE

Abstract— We study a networked control architecture forlinear time-invariant plants in which an unreliable data-rate
limited network is placed between the controller and the plant
input. The distinguishing aspect of the situation at hand is thatan unreliable data-rate limited network is placed between controller and the plant input. To achieve robustness with respect
to dropouts, the controller transmits data packets containing
plant input predictions, which minimize a finite horizon costfunction. In our formulation, we design sparse packets for ratelimited networks, by adopting an an ℓ0 optimization, which canbe effectively solved by an orthogonal matching pursuit method.
Our formulation ensures asymptotic stability of the control
loop in the presence of bounded packet dropouts. Simulation
results indicate that the proposed controller provides sparsecontrol packets, thereby giving bit-rate reductions for the caseof memoryless scalar coding schemes when compared to theuse of, more common, quadratic cost functions, as in linear
quadratic (LQ) control.

I. I NTRODUCTION
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]–[3] Many interesting
challenges arise and successful NCS design methods needto consider both control and communication aspects. In
particular, so-called packetized predictive control (PPC) has
been shown to have favorable stability and performance
properties, especially in the presence of packet-dropouts[4]–[10]. In PPC, the controller output is obtained throughsolving a finite-horizon cost function on-line in a receding
horizon manner. Each control packet contains a sequenceof 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 actuatorside, 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 givedesirable closed loop properties, the more unreliable the
network is, the larger the horizon length (and thus the numberof tentative plant input values contained in each packet)
Masaaki Nagahara is with the Graduate School of Informatics, Kyoto University, Kyoto, 606-8501, Japan; e-mail: nagahara@ieee.org.
Daniel Quevedo is with the School of Electrical Engineering & Computer
Science, The University of Newcastle, NSW 2308, Australia; e-mail:dquevedo@ieee.org. Jan Østergaard is with the Department of Electronic Systems, Aalborg University, Denmark; e-mail: janoe@ieee.org
This research was supported in part under the MEXT (Japan) Grant-inAid for Young Scientists (B) No. 22760317, and also Australian ResearchCouncil’s Discovery Projects funding scheme (project number DP0988601).

needs to be chosen. Clearly, in principle, this would require
increasing the network bandwidth (i.e., its bit-rate), unlessthe transmitted signals are suitably compressed.
To address the compression issue mentioned above, in thepresent work we investigate the use of sparsity-promoting
optimizations for PPC. Such techniques have been widely
studied in the recent signal processing literature in thecontext of compressed sensing (aka compressive sampling)[11]–[16]. The aim of compressed sensing is to reconstruct
a signal from a small set of linear combinations of the signal
by assuming that the original signal is sparse. The core idea
used in this area is to introduce a sparsity index in the
optimization. To be more specific, the sparsity index of avector v is defined by the amount of nonzero elements inv and is usually denoted by kvk0 , called the “ℓ0 norm.”
The compressed sensing problem is then formulated byan ℓ0 -norm optimization, which, being combinatorial is, inprinciple hard to solve [17]. Since sparse vectors containmany 0-valued elements, they can be easily compressed byonly coding a few nonzero values and their locations. A wellknown example of this kind of sparsity-inducing compressionis JPEG [18].
The purpose of this work is to adapt sparsity concepts foruse in NCSs over erasure channels. A key difference between
standard compressed sensing applications and NCSs is thatthe latter operate in closed loop. Thus, time-delays need to
be avoided and stability issues studied, see also [19]. To keep
time-delays bounded, we adopt an iterative greedy algorithm
called Orthogonal Matching Pursuit (OMP) [20], [21] forthe on-line design of control packets. The algorithm is very
simple and known to be dramatically faster than exhaustive
search. In relation to stability in the presence of boundedpacket-dropouts, our results show how to design the costfunction to ensure asymptotic stability of the NCS.
Our present manuscript complements our recent conference contribution [19], which adopted an ℓ1 -regularized ℓ2
optimization for PPC. A limitation of the approach in [19]is that for open-loop unstable systems, asymptotic stability
cannot be obtained in the presence of bounded packetdropouts; the best one can hope for is practical stability. Ourcurrent paper also complements the extended abstract [22],by considering bit-rate issues and also presenting a detailed
technical analysis of the scheme, including proofs of results.
To the best of our knowledge, the only other publishedworks which deal with sparsity and compressed sensing forcontrol are [23] which studies compressive sensing for state

v(k)
u(k)

u(x(k))
Controller

Fig. 1.

Bufferx(k)Plant

NCS with PPC. The dotted line indicates an erasure channel.

reconstruction in feedback systems, and [19], [24] which
focus on sampling and command generation for remoteapplications.
The remainder of this work is organized as follows:
Section II revises basic elements of packetized predictive
control. In Section III, we formulate the design of thesparse control packets in PPC based on sparsity-promoting
optimization. In Section IV, we study stability of the resultant
networked control system. Based on this, in Section V
we propose relaxation methods to compute sparse control
packets which leads to asymptotic (or practical) stability. A
numerical example is included in Section VI. Section VII
draws conclusions.
Notation:: We write N0 for {0, 1, 2, 3, . . .}, | · | refers tomodulus of a number. The identity matrix (of appropriate
dimensions) is denoted via I. For a matrix (or a vector) A,A⊤ denotes the transpose. For a vector v = [v1 , . . . , vn ]⊤ ∈Rn and a positive definite matrix P > 0, we definenX√|vi |, kvk∞ := max |vi |kvkP := v ⊤ P v, kvk1 :=i=1

i=1,...,n√and also denote kvk2 := v ⊤ v. For any matrix P , λmax (P )and λmin (P ) denote the maximum and the minimum eigen2
values of P , respectively; σmax
(P ) := λmax (P ⊤ P ).
II. PACKETIZED P REDICTIVE N ETWORKED C ONTROL
We consider discrete-time (LTI) plants with a scalar input:x(k + 1) = Ax(k) + Bu(k) + v(k),x(0) = x0 ,k ∈ N0 ,(1)

where x(k) ∈ Rn , u(k) ∈ R and v(k) ∈ Rn is the plantnoise. Throughout this work, we assume that the pair (A, B)
is reachable.
We are interested in an NCS architecture where thecontroller communicates with the plant actuator throughan erasure channel, see Fig. 1. This channel introduces
packet-dropouts, which we model via the dropout sequence{d(k)}k∈N0 in:
1, if packet-dropout occurs at instant k,d(k) ,
0, if packet-dropout does not occur at time k.
With PPC, as described, for instance, in [8], at each time
instant k, the controller uses the state x(k) of the plant (1)to calculate and send a control packet of the form
⊤


 
u x(k) , u0 x(k) , u1 x(k) , . . . , uN −1 x(k) ∈ RN
(2)
to the plant input node.

To achieve robustness against packet dropouts, buffering
is used. More precisely, suppose that at time instant k, wehave d(k) = 0, i.e., the data packet u x(k) is successfullyreceived at the plant input side. Then, this packet is stored ina buffer, overwriting
its previous contents. If the next packetu x(k + 1) is dropped, then the plant input u(k + 1) is set
to u1 x(k) , the second element of u x(k) . The elementsof u x(k)  are then successively used until some packet
u x(k + ℓ) , ℓ ≥ 2 is successfully received.
III. D ESIGN OF S PARSE C ONTROL PACKETSIn PPC discussed above, the control packet u x(k) istransmitted at each time k ∈ N0 through an erasure channel(see Fig. 1). It is often the case that the bandwidth of thechannel is limited, and hence one has to compress control
packets to a smaller data size, see also [25]. To design
packets which are easily compressible, we adapt techniques
used in the context of compressed sensing [11], [12] to design
sparse control vectors u x(k) . Since sparse vectors containmany 0-valued elements, they can be highly compressed byonly coding their few nonzero components and locations, as
will be illustrated in Section VI. Thus, the control objective
in this paper is to find sparse control packets u x(k)which ensure that the NCS with bounded packet dropouts
is asymptotically stable.
We define the sparsity of a vector u by its ℓ0 “norm,”
kuk0 , the amount of nonzero elements in u ∈ RN
and introduce the following sparsity-promoting optimization:u(x) , arg min kuk0u∈RN

subject to kx′N k2P +

N
−1
X
i=1

(3)
kx′i k2Q ≤ x⊤ W x,

where we omit the dependence on k, and
x′0 = x, x′i+1 = Ax′i + Bu′i ,⊤
u = u′0 , u′1 , . . . , u′N −1

i = 0, 1, . . . , N − 1,are plant state and input predictions. The matrices P > 0,
Q > 0, and W > 0 are chosen such that the feedback system
is asymptotically stable. The procedure of choosing thesematrices is presented in Section IV.
At each time instant k ∈ N0 , the controller uses the current
state x(k) to solve the above optimization with x = x(k)
thus providing the optimal control packet u x(k) . This
(possibly sparse) packet can be effectively compressed before
it is transmitted to the buffer at the plant side.
IV. S TABILITY A NALYSIS
In this section, we show that if• v(k) = 0,
• the matrices P , Q, and W in the proposed optimization(3) or (5) are appropriately chosen,• and the maximum number of consecutive dropouts isbounded,then the NCS is asymptotically stable. The proof is omitted
due to limitation of space.
To consider the stability of the networked system affectedby packet dropouts, we follow akin to what was done in[8] and denote the time instants where there are no packetdropouts, i.e., where d(k) = 0, as
K = {ki }i∈N0 ⊆ N0 ,ki+1 > ki , ∀i ∈ N0

whereas the number of consecutive packet-dropouts is denoted via:
mi , ki+1 − ki − 1,i ∈ N0 .

3) Compute constants ρ ∈ [0, 1) and c > 0 via⊤−1c1 , max λmax Φ⊤> 0,i P Φi (G G)
i=0,...,N −1

ρ , 1 − λmin (QP −1 ),4) Choose E such that 0 < E < (1 − ρ)P/c.
5) Compute W ⋆ = P − Q and set W := W ⋆ + E.
Then the sparse control packets u x(k) , k ∈ N0 , whichis the solution of the optimization (3) or (5) with theabove matrices, lead to asymptotic stability of the networked
control system.

(4)

Note that mi ≥ 0, with equality if and only if no dropouts
occur between instants ki and ki+1 .
When packets are lost, the control system unavoidably
operates in open-loop. Thus, to ensure desirable propertiesof the networked control system, one would like the numberof consecutive packet-dropouts to be bounded. In particular,
to establish asymptotic stability, we make the following
assumption: 1
Assumption 4.1 (Packet-dropouts): The number of consecutive packet-dropouts is uniformly bounded by the prediction horizon minus one, that is, mi ≤ N − 1, ∀i ∈ N0 .
We also assume that the first control packet u x(0) is
successfully transmitted, that is, m0 = 0.

Theorem 4.2 stated below shows how to design the matrices P , Q, and W in (3) to ensure asymptotic stability of the
networked control system in the presence of bounded packet
dropouts. Before proceeding, we introduce the matrices:




B
0
... 0
Φ0
 AB
 Φ1 
B... 0Φ,, = . 

..
..
.
.
.
.
.



.
.
.
.
. 
N −1
N −2
A
B A
B ... BΦN −1 iΦi , A B . . . B 0 . . . 0 , i = 0, 1, . . . , N − 1,A
 A2 Υ ,  .  , Q̄ , blockdiag{Q, . . . , Q, P },| {z }
 .. N −1
NA
which allow us to re-write (3) in vector form via
u(x) = arg min kuk0 subject to kGu − Hxk22 ≤ x⊤ W x,
u∈RN

(5)
where G , Q̄1/2 Φ and H , −Q̄1/2 Υ.
Theorem 4.2 (Asymptotic Stability): Suppose
that
Assumption 4.1 holds and that the matrices P , Q,
and W are chosen by the following procedure:
1) Choose Q > 0 arbitrarily.
2) Solve the following Riccati equation to obtain P > 0:
P = A⊤ P A − A⊤ P B(B ⊤ P B)−1 B ⊤ P A + Q.
1 If only stochastic properties are sought, then more relaxed assumptions
can be used, see related work in [25].

c , (1 − ρ)−1 1 − ρN c1 .

V. O PTIMIZATION VIA OMP
In this section, we consider the optimization(P0 ) :

min kuk0 subject to kGu−Hxk22 ≤ x⊤ W x.

u∈RN

The optimization (P0 ) is in general extremely complex since
it requires a combinatorial search that explores all possible
sparse supports of u ∈ RN . In fact, it is proved to be NP hard[17]. For such problem, there have been proposed alternativealgorithms that are much more tractable than exhaustive
search; see, e.g., the books [14]–[16].
One approach to the combinatorial optimization is an iterative greedy algorithm called Orthogonal Matching Pursuit(OMP) [20], [21]. The algorithm is very simple and dramatically faster than the exhaustive search. In fact, assumingthat G ∈ Rm×n and the solution u∗ of (P0 ) satisfiesku∗ k0 = k0 , then the OMP algorithm requires O(k0 mn)
operations, while exhaustive search requires O(mnk0 k02 )[26].2 The OMP algorithm for our control problem is shownin Algorithm 1. In this algorithm, supp{x} is the support
set of a vector x = [x1 , x2 , . . . , xn ]⊤ , that is, supp{x} ={i : xi 6= 0}, and g j denotes the j-th column of the matrix
G.
Next, we study stability of the NCS with control packets
computed by Algorithm 1.
Since Algorithm 1 always returns a feasible solution for(P0 ), we have the following result based on Theorem 4.2.
Theorem 5.1: Suppose that Assumption 4.1 holds and thatthe matrices P , Q, and W are chosen according to theprocedure given in Theorem 4.2. Then, the control packets
uOMP (x(k)), k ∈ N0 obtained by the OMP Algorithm 1
provide an asymptotically stable NCS.
Consequently, when compared to the method used in [19],Algorithm 1 has the following main advantages:• it is simple and fast,• it returns control packets that asymptotically stabilize
the networked control system
We note that in conventional transform based compression
methods e.g., JPEG, the encoder maps the source signal intoa domain where the majority of the transform coefficientsare approximately zero and only few coefficients carry significant information. One therefore only needs to encode
the few significant transform coefficients as well as their
2 For control applications, OMP has recently been proposed for use information control in [27].

supp{u}=S[k+1]r[k + 1] := Hx − Gu[k + 1].
k := k + 1.
end while
return u(x) = u[k].

locations. In our case, on the other hand, we use the OMP
algorithm to sparsify the control signal in its original domain,which simplifies the decoder operations at the plant side. To
obtain a practical scheme for closed loop control, we employ
memoryless entropy-constrained scalar quantization of the
non-zero coefficients of the sparse control signal and, in
addition, send information about the coefficient locations. We
then show, through computer simulations, that a significant
bit-rate reduction is possible compared to when performing
memoryless entropy-constrained scalar quantization of thecontrol signal obtained by solving the standard quadratic
control problem for PPC as in [4].3
VI. S IMULATION S TUDIES

45

40

35

30

25
0
10

(6)

This model is a constant-speed approximation of some of thelinealized dynamics of a Cessna Citation 500 aircraft, when
it is cruising at an altitude of 5000 (m) and a speed of 128.2
3 It is interesting to note that our proposed sparsifying controller could
also be useful in applications where there is a setup cost of the type found,for example, in inventory control; see, e.g., [28]. In such a case, it would
be advantageous to have many zero control values.

1

2

10

10

3

ν

10

4

10

5

10

Fig. 2. Regularization parameters νi versus control performance kxk2 for(Q1 ) (solid) and (Q2 ) (dash). The circles show the chosen parameters ν1
and ν2 .

(m/sec) [29, Section 2.6]. To obtain a discrete-time model,
we discretize (6) by the zero-order hold with sampling time
Ts = 0.5 (sec)4 . We set the horizon length (or the packetsize) to N = 10. We choose the weighting matrix Q in (3) as
Q = I, and choose the matrix W according to the procedureshown in Theorem 4.2 with E = 32 (1 − ρ)P/c < (1 − ρ)P/c.
A. Sparsity and Asymptotic Stability
We first simulate the NCS in the noise-free case wherev(k) = 0. We consider the proposed method using the OMP
algorithm and also the ℓ1 /ℓ2 optimization of [19]:(Q1 ) :

1min ν1 kuk1 + kGu − Hxk22 ,N2u∈Rwhere ν1 is a positive constant. To compare these twosparsity-promoting methods with traditional PPC approaches,
we also consider a finite-horizon quadratic cost function(Q2 ) :To assess the effectiveness of the proposed method, we
consider the following continuous-time plant model:ẋc = Ac xc + Bc u,−1.282200.98 0

0
0
1 0 
,
Ac = 
 −5.4293
0 −1.8366 0 −128.2 128.2
0 0


−0.3

0 
.

Bc = 
−17 
0Reg. Parameter vs Control Performance50||x||2 (dB)

Algorithm 1 OMP for sparse control vector u(x)Require: x ∈ Rn {observed state vector}
Ensure: u(x) {sparse control packet}
k := 0.
u[0] := 0.
r[0] := Hx − Gu[0] = Hx.
S[0] := supp{x[0]} = ∅.
while kr[k]k22 > x⊤ W x do
for j = 1 to N dog⊤j r[k]zj :== arg min kgj z − r[k]k22 .
kg j k22z∈R
ej := kg j zj − r[k]k22 .
end forFind a minimizer j0 6∈ S[k] such that ej0 ≤ ej , for all
j 6∈ S[k].
S[k + 1] := S[k] ∪ {j0 }u[k + 1] :=arg minkGu − Hxk22 .

min

u∈RN

1
ν2
kuk22 + kGu − Hxk22 ,22where ν2 is a positive constant, yielding the ℓ2 -optimal
controlu2 (x) = (ν2 I + G⊤ G)−1 G⊤ Hx.
To choose the regularization parameters ν1 in (Q1 ) and ν2in (Q2 ), we empirically compute the relation between eachparameter and the control performance, as measured by theℓ2 norm of the state {x(k)}99
k=0 . Fig. 2 shows this relation.
By this figure, we first find the optimal parameter for ν2 > 0that optimizes the control performance, i.e., ν2 = 3.1 × 102 .
Then, we seek ν1 that gives the same control performance,
namely, ν1 = 5.3 × 103 . Furthermore, we also investigate
the ideal least-squares solution u⋆ (x) that minimizes kGu−
Hxk2 .
With these parameters, we run 500 simulations withrandomly generated (Markovian) packet-dropouts that satisfy Assumption 4.1, and with initial vector x0 in which
4 This is done by MATLAB command c2d.

Sparsity ||u||[log plot] 2−norm of the state x(k)

0

14

0

OMP
Ideal
L2
L1/L2 (i)L1/L2 (ii)log10 ||x(k)||2

12

10

10

OMP
Ideal
L2
L1/L2 (i)L1/L2 (ii)

−20

10

−40

10

8

10

20

30

40

||u||0

0

50
k

60

70

80

90

100

[linear plot] 2−norm of the state x(k)

6
5

OMP
Ideal
L2
L1/L2 (i)L1/L2 (ii)

4
4
2

3

||x(k)||

2

2
1
0

0
0

10

20

30

40

50
k

60

70

80

90

100

Fig. 3. Sparsity of the control vectors by ideal (star), ℓ2 with regularization
parameter ν2 = 3.1×102 , OMP (solid), ℓ1 /ℓ2 with ν1 = 5.3×103 (dashdot) and ν1 = 5.3 (dash).

0

10

20

30

40

50
k

60

70

80

90

100

Fig. 4. 2-norm of the state x(k) for the four PPC designs: log plot (above)and linear plot (below).
Computational time

B. Bit-rate Issues
We next investigate bit-rate aspects for a Gaussian plantnoise process v(k). To keep the encoder and decoder simple,
we will be using memoryless entropy-constrained uniform
scalar quantization; see [30]. Thus, the non-zero elements ofthe control vector are independently encoded using a scalar
uniform quantizer followed by a scalar entropy coder. In
the simulations, we choose the step size of the quantizer
to be ∆ = 0.001, which results in negligible quantization
distortion. We first run 1000 simulations with 100 time
steps and use the obtained control vectors for designing

OMP
Ideal
L2
L1/L2 (i)L1/L2 (ii)

−2

10

−3

10
Time (sec)each element is independently sampled from the normal
distribution with mean 0 and variance 1. Fig. 3 shows theaveraged sparsity of the obtained control vectors. The ℓ1 /ℓ2
optimization with ν1 = 5.3 × 103 always produces muchsparser control vectors than those by OMP. This property
depends on how to choose the regularization parameterν1 > 0. In fact, if we choose smaller ν1 = 5.3, the sparsity
changes as shown in Fig. 3. On the other hand, if we use asufficiently large ν1 > 0, then the control vector becomes
0. This is indeed the sparsest control, but leads to very
poor control performance: the state diverges until the control
vector becomes nonzero (see [19]).
Fig. 4 shows the averaged 2-norm of the state x(k) as afunction of k for all 5 designs. We see that, with exceptionof the ℓ1 /ℓ2 optimization based PPC, the NCSs are nearly
exponentially stable. In contrast, if the ℓ1 /ℓ2 optimization of[19] is used, then only practical stability is observed. The
simulation results are consistent with Corollary 5.1 and ourprevious results in [19]. Note that the ℓ1 /ℓ2 optimization with
ν1 = 5.3 shows better performance than that with ν1 = 5.3×
103 , while ν1 = 5.3 × 103 gives a much sparser vector. Thisshows a tradeoff between the performance and the sparsity.
Fig. 5 shows the associated computation times. The ℓ1 /ℓ2
optimization is faster than OMP in many cases. Note thatthe ideal and the ℓ2 optimizations are much faster, since theyrequire just one matrix-vector multiplication.

−4

10

−5

10

−6

10

0

10

20

30

Fig. 5.

40

50
k

60

70

80

90

100

Computational time.

entropy coders. A separate entropy coder is designed for eachelement in the control vector. For the first N/2 elements inthe vector, we always use a quantizer followed by entropy
coding. For the remaining N/2 elements, we only quantize
and entropy code the non-zero elements. We then sendadditional N/2 bits indicating, which of the N/2 elements
have been encoded. The total bit-rate for each control vector
is obtained as the sum of the codeword lengths for eachindividual non-zero codeword +N/2 bits. For comparison,
we use the same scalar quantizer with step size ∆ = 0.001
and design entropy coders on the data obtained from theℓ2 optimization. Since the control vectors in this case are
non-sparse, we separately encode all N elements and sum
the lengths of the individual codewords to obtain the total
bit-rate. In both of the above cases, the entropy coders
are Huffman coders. Moreover, the system parameters are
initialized with different random seeds for the training andtest situations, respectively. The average rate per control
vector for the OMP case is 55.36 bits, whereas the averagerate for the ℓ2 case is 112.57 bits. Thus, due to sparsity, a50.8 percent bit-rate reduction is on average achieved.

2−norm of the state x(k)210||x(k)||2

OMP
L1/L2
Ideal
L2

1

10

0

10

0

10

Fig. 6.

20

30

40

50
k

60

70

80

90

100

2-norm of the state x(k) with Gaussian noise.
Sparsity ||u||0

10
OMP
L1/L2

9

8

7

||u||0

6

5

4
3

2

1

0

0

Fig. 7.

10

20

30

40

50
k

60

70

80

90

100

Sparsity of the control vectors with Gaussian noise.

Fig. 6 shows the 2-norm of the state x(k) and Fig. 7
shows the sparsity. We can also see the tradeoff between theperformance and the sparsity in this case.
VII. C ONCLUSION
We have studied a packetized predictive control formulation with a sparsity-promoting cost function for error-prone
rate-limited networked control system. We have given sufficient conditions for asymptotic stability when the controller
is used over a network with bounded packet dropouts. Simulation results indicate that the proposed controller provides
sparse control packets, thereby giving bit-rate reductions
when compared to the use of, more common, quadratic
cost functions. Future work may include further study of
performance aspects and the effect of plant disturbances.
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