Sparse Representations for Packetized Predictive
Networked Control 
Masaaki Nagahara ∗ Daniel E. Quevedo ∗∗∗Graduate School of Informatics, Kyoto University, Kyoto, 606-8501,
JAPAN (e-mail: nagahara@ieee.org).
∗∗
School of Electrical Engineering & Computer Science,
The University of Newcastle, Callaghan, NSW 2308,
AUSTRALIA (e-mail: dquevedo@ieee.org).

Abstract: We investigate a networked control architecture for LTI plant models with a scalar input.
Communication from controller to actuator is over an unreliable network which introduces packet
dropouts. To achieve robustness against dropouts, we adopt a packetized predictive control paradigm
wherein each control packet transmitted contains tentative future plant input values. The novelty of ourapproach is that we seek that the control packets transmitted be sparse. For that purpose, we adapt tools
from the area of compressed sensing and propose to design the control packets via on-line minimizationof a suitable 1/2 cost function. We then show how to choose parameters of the cost function to ensurethat the resultant closed loop system be practically stable, provided the maximum number of consecutive
packet dropouts is bounded. A numerical example illustrates that sparsity reduces bit-rates, thereby
making our proposal suited to control over unreliable and bit-rate limited networks.
Keywords: networked control, model predictive control, sparse representation, 1/2 optimization.
1. INTRODUCTION
Compressed sensing, which is a hot topic in signal processing,aims at reconstructing signals by assuming that the original
signal is sparse; see, e.g.,[Candes,Wakin (2008); ZibulevskyElad (2010)]. The core idea used in this area is to introduce
a sparsity index in the optimization used for decoding. To be
more specific, the sparsity index of a vector v is defined by
the amount of nonzero elements in v and is usually denotedby v0 , called the “0 norm.” The compressed sensing decoding problem is then formulated by optimization with 0 norm regularization. The associated optimization problem ishowever hard to solve, since it is a combinatorial one. Thus, it
is common to introduce a convex relaxation by replacing the 0norm with the 1 norm. Under some assumptions, the solutionof this relaxed optimization is known to be exactly the same asthat of the 0 -norm optimization [Candes,Wakin (2008)]. Thatis, by minimizing the 1 -norm, one can obtain a sparse solution.

unreliable the network is, the larger the horizon length (and thus
the number of tentative plant input values contained in eachpacket) should be. Therefore, to encompass bit-rate limitationsof digital networks, it becomes natural to seek that the control
packets provided by PPC be sparse.
In order to obtain sparse control packets, we propose to design
the latter by minimization of an 1/2 cost function, whichallows one to trade control performance for sparsity of thepackets. The associated optimization can be effectively solved
by iteration methods, as in compressed sensing applications,see also [Daubechies et al. (2004); Beck-Teboulle (2009)],
and is, thus, suitable for practical control implementations. We
show how to choose the parameters of the cost function toachieve practical stability of the closed loop in the presenceof bounded packet dropouts. We then illustrate that sparsity
may reduce bit-rates. This makes the proposed control method
suitable for situations where the network is not only unreliable,
but also bit-rate limited.

The purpose of the present work is to use sparsity-inducing
techniques in the context of controller design for networked
control applications. In particular, we will focus on packetized
predictive control (PPC); see, e.g., [Bemporad (1998); Casavola
et al. (2006); Tang-Silva (2007); Quevedo-Nešić (2011)]. As
in regular model predictive control (MPC) formulations, a costfunction is used in PPC to design the controller output. Eachcontrol packet contains a sequence of tentative plant inputs
for a finite horizon of future time instants. Packets which arereceived at the plant actuator side, are stored in a buffer to be
used whenever later packets are dropped by the network. When
there are no dropouts, PPC reduces to model predictive control.
For PPC to give desirable closed loop properties, the more

Before proceeding, we note that only few works on MPC (andnone on PPC) have explicitly studied the use of cost functions
with 1 norms [Keerthi-Gilbert (1988); Lazar et al. (2006)]. In
fact, most MPC formulations use quadratic cost functions (seee.g., [Goodwin et al. (2005); Maciejowski (2002); RawlingsMayne (2009)]). To the best of our knowledge, no specificresults on stability of mixed 1/2 MPC (or the, more general,
PPC) have been documented. We also note that our approachfor obtaining sparsity differs somewhat from that used in compressed sensing. In fact, our objective is to derive sparse signals
for efficient encoding, whereas compressed sensing aims at
decoding sparse signals.

 The research was supported in part under Australian Research Council’s

The remainder of this work is organized as follows: Section 2
revises basic elements of packetized predictive control. In Sec-

Discovery Projects funding scheme (project number DP0988601).

tion 3, we show how to choose the cost function to obtain sparsecontrol packets. In Section 4, we study stability of the resultant
networked control system. A numerical example is included in
Section 5. Section 6 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 definen√vP := v  P v, v1 :=|vi |, v∞ := max |vi |√

i=1

i=1,...,nand also denote v2 := v  v. For any matrix P , λmax (P )and λmin (P ) denote the maximum and the minimum eigenval2
ues of P , respectively. We also define σmax
(P ) := λmax (P  P ).
2. PACKETIZED PREDICTIVE NETWORKED CONTROL
We consider the following discrete-time linear and timeinvariant plant model:x(k + 1) = Ax(k) + Bu(k), k ∈ N0 , x(0) = x0 , (1)where x(k) ∈ Rn and u(k) ∈ R for k ∈ N0 . We assume thatthe realization (A, B) is reachable.
We are interested in a networked control architecture where
the controller communicates with the plant actuator throughan unreliable 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 instant k.
In packetized predictive control (PPC), as described, e.g., in[Quevedo-Nešić (2011); Quevedo et al. (2011)], 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

Controller

U (x(k))To achieve robustness against packet dropouts, buffering isused. To be more specific, suppose that at time instant k, wehave d(k) = 0, i.e, the data packet U (x(k)) defined in (2) is
successfully received at the plant input side. Then, this packet isstored in a buffer, overwriting its previous contents. If the next
packet U (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 is
successfully received, i.e., no dropout occurs. More formally,
the sequence of buffer states, say {b(k)}k∈N0 , satisfies therecursionb(k) = d(k)Sb(k − 1) + (1 − d(k))U (x(k)),(3)
where b(0) = 0 ∈ RN and with⎤⎡
0 1 0 ... 0⎢ .. . . . . . . .. ⎥⎢. . . . .⎥
N ×N
⎥
.
S⎢
⎢0 . . . 0 1 0⎥ ∈ R⎣0 . . . . . . 0 1⎦
0 ......... 0
The buffer states ultimately give rise to the plant inputs in (3)viau(k) = [1 0 . . . 0] b(k).

u(k)

Plant

x(k)

Fig. 1. Networked Control System with PPC. The dotted line
indicates an erasure channel.
3. SPARSE CONTROL PACKET DESIGNAs foreshadowed in the introduction, in the present work weseek that the control packets {U (x(k))}k∈N0 be sparse. For that
purpose, we propose to use a dynamic 1/2 optimization. Moreprecisely, at each time instant k, the controller minimizes thefollowing cost function:
N
−1
N
−1


2
2
J(U, x(k))  x(N |k)P +x(i|k)Q + μ|ui |, (4)
i=0

i=0


where U = [u0 , u1 , . . . , uN −1 ] . In (4), {x(i|k)}, i ∈{0, 1, . . . , N } are predicted plant states, which are calculatedby x(i + 1|k) = Ax(i|k) + Bui , i = 0, 1, . . . , N − 1 withx(0|k) = x(k), the observed state of the plant (1) at time instantk. The parameters P > 0, Q > 0, and μ > 0 allow the designer
to trade control performance for control effort and sparsity of
the control packets. As we will see in Section 4, the choice ofdesign parameters also influences closed loop stability of theresultant networked control system.
If we introduce the following matrices:
⎡
⎤
B
0
... 0
B
... 0⎥
⎢ AB
Φ⎢
,
.
.
. . .. ⎥
⎣
..
..
. .⎦
AN −1 B AN −2 B . . . B

⎡

⎤
A
2
⎢A ⎥
⎢
⎥
Υ  ⎢ .. ⎥ ,
⎣ . ⎦
AN

Q̄  blockdiag{Q, . . . , Q, P },N −1U (x(k))  [u0 (x(k)) u1 (x(k)) . . . uN −1 (x(k))] ∈ RN
(2)
to the plant input node.

Bufferthen the cost function in (4) can be re-written in vector form viaJ(U, x(k)) = GU − Hx(k)22 + μU 1 + x(k)2Q , (5)where G  Q̄1/2 Φ, H  −Q̄1/2 Υ. Consequently, theoptimal U (x(k)) minimizing (5) can be numerically obtainedby the following iteration [Zibulevsky-Elad (2010)]:U (x(k))  arg min J(U, x(k)) = lim Uj ,
U

where



j→∞

Uj+1 = S2μ/c1 
G (Hx(k) − GUj ) + Uj , j ∈ N0 , (6)
c

with

⎡

⎤
sgn(v1 ) max{0, |v1 | − 2μ/c}⎥⎢..
S2μ/c (v)  ⎣
⎦,
.

(7)

sgn(vm ) max{0, |vm | − 2μ/c}

see Fig. 2. The constant c in (6) is chosen to satisfy c >λmax (G G), in which case the iteration (6) converges to the
optimizer of (5) for any initial value U0 ∈ RN [Daubechies
et al. (2004)]. The convergence rate of this iteration is known tobe O(1/j). Faster methods have been proposed, e.g., in [BeckTeboulle (2009)].
Remark 1. The difference between the cost function in (4) andmost MPC formulations is the penalty on the input vector

Lemma 3. (Bounds of V (x)). For any x ∈ Rn , we havez

λmin (Q)x22 ≤ V (x) ≤ φ(x2 )
where

−2μ/c

φ(x2 )  a1 x2 + (a2 + λmax (Q))x22 ,
(10)
 + 


√
2
a1  μ n σmax G H , a2  σmax
(GG+ − I)H ,v
0

G+  (G G)−1 G .

2μ/c

Fig. 2. Function z = sgn(v) max{0, |v| − 2μ/c}, v ∈ R, usedin (7).
U . Standard MPC uses an 2 penalty U 2R , R > 0 for
attenuating the control U (see e.g., [Maciejowski (2002)]). In
our formulation, we consider an 1 penalty U 1 , which is
introduced in order to obtain a sparse representation of thecontrol packet U (x(k)).

4. STABILITY ANALYSIS OF 1/2 PPC
We will next analyze closed loop stability of 1/2 PPC, aspresented in Section 3, with bounded packet dropouts. Our
analysis uses elements of the technique introduced in [Quevedo
et al. (2007)] and which was later refined in [Quevedo-Nešić
(2011)].
A distinguishing aspect of the situation at hand is that, for openloop unstable plants, even when there are no packet dropouts,
asymptotic stability will not be achieved, despite the fact thatthe plant-model in (1) is disturbance-free. This can be easily
shown by considering the iterationin (6), with U0 = 0and
a plant state x(k) ∈ Ω  x ∈ Rn : G Hx∞ ≤ 2μ . It
follows directly from (6) that, in this case, limj→∞ Uj = 0.
Since this limit value is independent of the initial value U0 , wehave x(k) ∈ Ω ⇒ U (x(k)) = 0. Thus, if x(k) ∈ Ω, and there
are no dropouts, then the control u(k) = 0. That is, the control
system (1) behaves as an open-loop system in the set Ω. Hence,
asymptotic stability will in general not be achieved, if A 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 study practicalstability (i.e., stability of a set) of the networked control system.
For that purpose, we will analyze the value function
V (x)  min J(U, x)U(8)and prepare the following two technical lemmas.:
Lemma 2. (Riccati equation). Let r > 0. Suppose P > 0 is thesolution to the Riccati equation
P = A P A − A P B(B  P B + r)−1 B  P A + Q,
and let K = −(B  P B + r)−1 B  P A. Then(9)

(A + BK) P (A + BK) − P + Q + rK  K = 0.
Proof. First, we assumed Q > 0 and hence (Q1/2 , A) isobservable. We also assumed that (A, B) is reachable. Thus,
for any r > 0 the Riccati equation has a unique solution P > 0[Zhou et al. (1996)]. Direct calculation gives the result.
Proof. First, since (A, B) is assumed to be reachable, we haveB = 0, and hence rank(Φ) = rank(diag(B)) = N . This andthe assumption that P > 0 and Q > 0 imply that G = Q̄1/2 Φ
has full column rank (i.e., rank(G) = N ). Therefore, G G isinvertible. Then consider a vector
U  (x)  G+ Hx = (G G)−1 G Hx.
(11)
Applying this vector to J(U, x) gives
J(U  (x), x) = μG+ Hx1 + (GG+ − I)Hx22 + x2Q√≤ μ nG+ Hx2 + (GG+ − I)Hx22 + x2Q√≤ μ nσmax G+ H x2
 2 


(GG+ − I)H + λmax (Q) x22 ,+ σmax√where we used the norm inequality v1 ≤ nv2 for anyv ∈ Rn [Bernstein (2005)]. This and the definition of V (x) in(8) provide the upper bound on V (k).
To obtain the lower bound, we simply note that by the definitionof J(U, x), we have x2Q ≤ J(U, x) for any U ∈ RN , and
hence λmin (Q)x22 ≤ V (x).
+Remark 4. The vector U (x) = G Hx used in (11) is theminimizer of the unconstrained cost function GU − Hx2 andalso approximates U (x), the optimizer of the 1/2 cost (4) for
plant state x. Since the following bound holds [Fuchs (2004)]:
G (GU (x) − Hx)∞ = G G(U (x) − U  (x))∞ ≤ μ,
the upper bound for V (x) given in Lemma 3 will be tight, if μ
is small.

Having established the above preliminary results, we introduce
the iterated mapping f i with implicit (open-loop optimal) input
U (x) = [u0 (x), . . . , uN −1 (x)] = arg min J(U, x)Ubyf i (x)  Ai x +i−1Ai−1−l Bul (x),i = 1, 2, . . . , N. (12)l=0This mapping describes the plant state evolution during periodsof consecutive packet dropouts. Note that, since the input U (x)is not a linear function of x (see Section 3, also Fig. 2), thefunction f i (x) is nonlinear. We have the following lemma:
Lemma 5. (Open-loop bound). Assume that P > 0 satisfies (9)withμ2,  > 0.
(13)
r=
4
n
Then for any x ∈ R , we have
V (f i (x)) − V (x) ≤ −λmin (Q)x22 + , i = 1, 2, . . . , N.
Proof. Fix i ∈ {1, . . . , N − 1} and consider the sequenceŨ = {ui (x), ui+1 (x), . . . , uN −1 (x), ũN , . . . , ũN +i−1 } ,where ũN +j (j = 0, 1, . . . , i − 1) is given byũN +j = K x̃N +j , x̃N +j+1 = Ax̃N +j + B ũN +j ,with K as in Lemma 2 and where x̃N = f N (x). We then havei

J(Ũ , f (x))= x̃N +i 2P +

N
−1




f l (x)2Q + μ|ul (x)|



l=i

+

N
+i−1



x̃l 2Q + μ|ũl |l=N= V (x) −i−1f l (x)2Q + μ|ul (x)|l=0+ x̃N +i 2P − f N (x)2P +

N
+i−1



x̃l 2Q + μ|ũl |l=N= V (x) −
+

i−1



f l (x)2Q + μ|ul (x)|



l=0
N
+i−1



x̃l+1 2P − x̃l 2P + x̃l 2Q + μ|ũl |l=N

By the relation x̃l+1 = (A + BK)x̃l and ũl = K x̃l for
l = N, N + 1, . . . , N + i − 1, and by Lemma 2, we can boundthe terms in the last sum above by
x̃l+1 2P − x̃l 2P + x̃l 2Q + μ|ũl |
μ2 N 
K K x̃l= x̃l (A + BK) P (A + BK) − P + Q +
4

2

μ2 N
2

≤ ,
−
+
|K x̃l | −
4
μN
N
N
Thus, the cost function J(Ũ , f i (x)) can be upper bounded by
i−1

 l

f (x)2Q + μ|ul (x)| + 
J(Ũ , f i (x)) ≤ V (x) −l=0≤ V (x) − x2Q + ≤ V (x) − λmin (Q)x22 + ,
(14)
where we have used the relation f 0 (x) = x. Since V (f i (x))
is the minimal value of J(U, f i (x)) among all U ’s in RN , wehave V (f i (x)) ≤ J(Ũ , f i (x)), and hence
V (f i (x)) ≤ V (x) − λmin (Q)x22 + .
For the case i = N , we consider the sequence Ũ =
N −1{ũN , ũN +1 , . . . , ũ2N −1 }. If we define l=N = 0, then (14)follows as in the case i ≤ N − 1.
The above result can be used to derive the following contraction
property of the optimal costs during periods of successive
packet dropouts:
Lemma 6. (Contracting property). Let  > 0. Assume that P >0 satisfies (9) with r as in (13). Then there exists a real numberρ ∈ (0, 1) such that for all x ∈ Rn , we haveN λmin (Q)V f i (x) ≤ ρV (x) +  +, i = 1, 2, . . . , N.
4
Proof. In this proof, we borrow a technique used in the proofof [(Lazar, 2009, Theorem 4.2.5)]. By Lemma 3, for x = 0 wehave 0 < V (x) ≤ a1 x2 + (a2 + λmax (Q))x22 .
Now suppose that 0 < x2 ≤ 1. Then x22 ≤ x2 and hence
V (x) ≤ (a1 + a2 + λmax (Q))x2 .

From Lemma 5, it follows that
V f i (x) ≤ V (x) − λmin (Q)x22 + λmin (Q)x2≤ 1−
V (x)−λmin (Q) x22 − x2 + 
V (x)


λmin (Q)λmin (Q)+≤ 1−
V (x) +
a1 + a2 + λmax (Q)4λmin (Q)= ρV (x) +
+ ,
4
with
λmin (Q)
.
(15)
ρ1−
a1 + a2 + λmax (Q)Since 0 < λmin (Q) ≤ λmax (Q), a1 > 0, and a2 > 0, it follows
that ρ ∈ (0, 1).
Next, consider the case where x2 > 1 so that x2 < x22
andV (x) < (a1 + a2 + λmax (Q))x22 .
This and Lemma 5 giveV f i (x) ≤ V (x) − λmin (Q)x22 + λmin (Q)x22= 1−
V (x) + V (x)λmin (Q)V (x) + 
< 1−
a1 + a2 + λmax (Q)λmin (Q)+ .
= ρV (x) +  ≤ ρV (x) +4
Finally, if x = 0, then the above inequality also holds since
V (0) = 0.

We will next use Lemma 6 to establish sufficient conditions forpractical stability of PPC in the presence of packet-dropouts.
To state our results, in the sequel we denote the time instants
where there are no packet-dropouts, 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 denotedvia:
mi  ki+1 − ki − 1, i ∈ N0 .
(16)
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 operates in openloop. Thus, to ensure desirable properties of the networked control system, one would like the number of consecutive packetdropouts to be bounded. In particular, to establish practicalstability, we make the following assumption: 1
Assumption 1. (Packet-dropouts). The number of consecutive
packet-dropouts is uniformly bounded by the prediction horizon minus one, i.e., we have mi ≤ N − 1, ∀i ∈ N0 .

Theorem 7 stated below shows how to design the parameters of
the cost function to ensure practical stability of the networked
control system in the presence of bounded packet dropouts.
Theorem 7. (Practical stability of 1/2 PPC). Suppose that Assumption 1 holds. Let  > 0 and choose P > 0 to satisfy (9)with r as in (13). Then for all k ∈ N0 we have
√ i+1 φ(x(k0 )2 )x(k)2 ≤ ( ρ)
+ Δ,
(17)
λmin (Q)
1 If only stochastic properties are sought, then more relaxed assumptions canbe used, see related work in [Quevedo et al. (2011)].

where i ∈ N0 is such that k ∈ {ki + 1, . . . , ki+1 },



N
ρ

+
Δ
1 − ρ λmin (Q)4Table 1. Control packets U (ki )(18)

ρ is given in (15), and φ(·) is as in (10).
Proof. Fix i ∈ N0 and note that at time instant ki , the control
packet is successfully transmitted to the buffer. Then until thenext packet is received at time ki+1 , mi consecutive packetdropouts occur. By the PPC strategy, the control input becomesu(ki + l) = ul (x(ki )), l = 1, 2, . . . , mi , and the states x(k),k = ki + 1, . . . , ki + mi , are determined by these open-loop
controls. Since, by assumption, we have mi ≤ N − 1, Lemma
6 givesN λmin (Q)V (x(k)) ≤ ρV (x(ki )) +  +(19)
4
for k ∈ {ki + 1, ki + 2, . . . , ki + mi }, and also for ki+1 = ki +mi + 1, we haveN λmin (Q).
(20)
V (x(ki+1 )) ≤ ρV (x(ki )) +  +4
Now by induction from (20), it is easy to see thatV (x(ki ))N λmin Qii−1≤ ρ V (x(k0 )) + (1 + ρ + · · · + ρ )  +
4


Q
1
N
λ
min
+
.
≤ ρi φ(x(k0 )2 ) +1−ρ
4This inequality and (19) give the boundρN λmin Q
V (x(k)) ≤ ρi+1 φ(x(k0 )2 ) ++1−ρ
4for k ∈ {ki + 1, ki + 2, . . . , ki+1 − 1}, and this inequality also
holds for k = ki+1 . Finally, by using the lower bound of V (x)provided in Lemma 3, we have
V (x(k))√ i+1 φ(x(k0 )2 )x(k)2 ≤≤ ( ρ)+ Δ,
λmin (Q)λmin (Q)√√√since a + b ≤ a + b, for all a, b ≥ 0.
Theorem 7 establishes practical stability of the networked control system. It shows that, provided the conditions are met, theplant state will be ultimately bounded in a ball of radius Δ.
It is worth noting that, as in other stability results which use
Lyapunov techniques, this bound will, in general, not be tight.
5. DESIGN EXAMPLES
To illustrate properties of the 1/2 PPC strategy proposed inthis work, we consider a plant model of the form (1) with 2⎡⎤
1.2597574 −0.265722 −0.6776537 1.1712147⎢ −0.0066489 −0.846387 −0.4174316 1.1930255 ⎥A=⎣,−0.4610984 −0.1307435 −0.1483141 0.2842062 ⎦−0.4855527 0.2480541 1.8002141 0.7398921⎡⎤
1.3372142⎢ −2.9903216 ⎥B=⎣.
0.9703207 ⎦−0.4056704
2 The elements of these matrices are generated by random sampling from thenormal distribution with mean 0 and variance 1. Note that the matrix A has 3
unstable eigenvalues and 1 stable eigenvalue.

1/2

2

i=0
- 2.6320.085- 2.21100- 2.632- 0.106- 1.869
0.102- 0.679i=10- 1.825- 0.022- 0.753
00.007- 1.74- 0.162- 0.7620.213

i=2- 1.8090- 0.82600- 1.733- 0.154- 0.778
0.207- 0.201i=3- 0.085- 0.8900.21
00- 0.137- 0.7590.169- 0.2130.229

i=4- 0.909
0
0.157
0.322
0
- 0.6510.292- 0.4650.150- 0.224

We set the horizon length in the cost function (4) to N = 5 andchoose weights μ = 100, Q = I and P as the solution to the
Riccati equation (9) with r = μ. In this case, the real number in Theorem 7 equals 25, computed by (13).
For comparison, in addition to the 1/2 PPC, we also synthesize PPC with a conventional 2 cost function, namely
J2 (U, x(k))  x(N |k)2P +

N
−1

i=0

x(i|k)2Q + μN−1|ui |2 .

i=0(21)
We first simulate a packetized predictive networked control
system setup as in Fig. 1. The number mi of consecutive packet
dropouts, see (16), is chosen from the uniform distribution on{1, 2, 3, 4}. The initial plant state is set to x(0) = [1, 1, 1, 1] .
Table 1 shows the first 5 successfully transmitted control packets U (ki ), i = 0, 1, 2, 3, 4 designed by the proposed method(1/2 ) and the quadratic one given in (21). It can be observed
that, for the present situation, the control packets provided bythe 1/2 design are more sparse than those obtained throughthe 2 formulation.
To study bit-rate aspects, we next quantize the packets {U (k)},by using 8-bit uniform quantization with step size 0.25 ineach component. Fig. 3 illustrates the 2 norm of the statex(k) obtained when using the 1/2 PPC and also with the2 PPC. Whilst both controllers give a loop which seems
practically stable and exhibits comparable performance, there isa difference in the quantized packets generated by each method.
This can be appreciated in Fig. 4, which shows a histogramof the quantized control values {ul (k)}, l ∈ {0, 1, . . . , N −1}, k ∈ {1, 2, . . . , 100} in all transmitted packets U (k) =[u0 (k), . . . , uN −1 (k)] . As indicated in Fig. 4, control packets
designed by the proposed method include much more zeros
than those obtained by the conventional one, see (21). In fact,
the packets obtained by 1/2 optimization include 307 zero
values, whereas the conventional ones contain only 218 zeros.
Since the total amount of transmitted packets is 100, in the 1/2
formulation about 3/5 of the transmitted signals are zero. Thisobservation suggests that the associated bit-rates of the signal
will be small.
To further investigate this issue, we compute the (discrete)entropyof the control values {ul (k)}, defined as H(u) − u P(u) log2 P(u) [bits], where P(u) is the probability
mass function of u; see, e.g., [Cover-Thomas (2006)]. The
function P(u) can be approximately estimated by the histogram
as in Fig. 4. Note that here we do not take account of the entropyof control vector U (k), that is, we assume scalar quantization.
In the situation studied with x(0) = [1, 1, 1, 1] , the entropy
by the 1/2 optimization proposed is 8.6177, while that by theconventional 2 approach is 9.5345. We next execute 10000

6. CONCLUSION

2 norm of the state x(k)
10
9
8
7

|x(k)|2

6
5
4
3
2
1
0
0

10

20

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k

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Fig. 3. 2 norm of the state x(k): proposed 1 /2 (solid) andconventional 2 (dots)

We have studied a packetized predictive control formulation
with an 1/2 cost function. The associated optimization can besolved effectively and is, thus, suitable for implementation ina real-time controller. We have given sufficient conditions forpractical stability when the controller is used over a networkwith bounded packet dropouts. Numerical results indicate thatthe proposed controller provides sparse control packets, thereby
giving bit-rate reductions when compared to the use of, morecommon, quadratic cost functions. Future work may include
the further study of performance aspects and the effect of plantdisturbances.
ACKNOWLEDGEMENT

Histogram

The authors would like to thank Jan Østergaard for his helpfulcomments.

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REFERENCES

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A. Beck and M. Teboulle (2009), A fast iterative shrinkage-thresholding algorithm for linear inverse problems, SIAM J. Imaging Sciences, vol. 2, No. 1,
pp. 183–202, 2009.
A. Bemporad (1998), Predictive control of teleoperated constrained systems
with unbounded communication delays, IEEE CDC., pp. 2133–2138.
D. S. Bernstein (2005), Matrix Mathematics, Princeton University Press, 2005.
E. J. Candes and M. B. Wakin (2008), An introduction to compressive sampling, IEEE Signal Processing Magazine, vol. 25, pp. 21–30.
A. Casavola, E. Mosca, and M. Papini, Predictive teleoperation of constrained
dynamic systems via Internet-like channels, IEEE Trans. Contr. Syst. Technol., vol. 14, no. 4, pp. 681–694.
T. M. Cover and J. A. Thomas (2006), Elements of Information Theory, 2ndEd., Wiley-International.
I. Daubechies, M. Defrise, and C. De-Mol (2004), An iterative thresholding algorithm for linear inverse problems with a sparsity constraint, Commun. Pure Appl. Math., vol. 57, no. 11, pp. 1413–1457.
J. Fuchs (2004), On sparse representation in arbitrary redundant bases, IEEE
Trans. Inform. Theory, vol. 50, no. 6, pp. 1341–1344.
G. C. Goodwin, M. M. Seron, and J. A. De Doná (2005), Constrained Control
and Estimation, Springer.
S. S. Keerthi and E. G. Gilbert (1988), Optimal infinite-horizon feedback
law for a general class of constrained discrete-time systems: stability andmoving-horizon approximations, Journal of Optimization Theory and Applications, Vol. 57, No. 2, pp. 265–293.
M. Lazar (2009), Predictive Control Algorithms for Nonlinear Systems, Doctoral Thesis, Technical University of Iasi, Romana.
M. Lazar, W. P. M. H. Heemels, S. Weiland, and A. Bemporad (2006),
Stabilizing model predictive control of hybrid systems, IEEE Trans. Automat. Contr., vol. 51, no. 11, pp. 1813–1818.
J. M. Maciejowski (2002), Predictive Control with Constraints, Pearson Education.
D. E. Quevedo and D. Nešić, Input-to-state stability of packetized predictive control over unreliable networks affected by packet-dropouts, IEEE
Trans. Automat. Contr., Vol. 56, No. 2, pp. 370–375, 2011.
D. E. Quevedo and J. Østergaard and D. Nešić, Packetized predictive control
of stochastic systems over bit-rate limited channels with packet loss, IEEE
Trans. Automat. Contr., vol. 56, No. 11, 2011.
D. E. Quevedo, E. I. Silva, and G. C. Goodwin (2007), Packetized predictive
control over erasure channels, Proc. Amer. Contr. Conf., pp. 1003–1008.
J. B. Rawlings and D. Q. Mayne (2009), Model Predictive Control Theory and
Design, Nob Hill Publishing.
P. L. Tang and C. W. de Silva (2007), Stability validation of a constrainedmodel predictive networked control system with future input buffering,
Int. J. Contr., Vol. 80, No. 12, pp. 1954–1970.
K. Zhou, J. C. Doyle, and K. Glover (1996), Robust and Optimal Control,
Prentice-Hall.
M. Zibulevsky and M. Elad (2010), L1-L2 optimization in signal and image
processing, IEEE Signal Processing Magazine, Vol. 27, pp. 76–88.

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50

0

−1

−0.5

0

0.5

1

Fig. 4. Histogram of the quantized values in all transmitted
packets: proposed 1/2 (left) and conventional 2 (right)

Sparsity

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13.6

||x||2

13.4
13.2
13
0

Fig. 5. Sparsity N − U 0 (top figure) and performance x2
(below), as a function of μ.
simulations with initial plant states x(0) whose elements are
randomly sampled from the Gaussian distribution with mean 0
and variance 1. The average entropies obtained are 12.2560 for1/2 optimization and 15.5701 for the 2 formulation. Since
the entropy of the signal transmitted serves as a measure of thecode length, in the cases studied, the 1/2 PPC method will
require lower bit-rates than PPC with an 2 cost function.
Fig. 5 illustrates the tradeoff between control performance andsparsity, which, as noted above, is related to bit-rates. The figure
illustrates the average sparsity 5 − U 0 , where100

U 0 1 
U (k)0 ,100
k=1

and the achieved performance x2 (the 2 norm of the state
sequence x), for μ ∈ [0, 100]. The other parameters are thesame as used above. As can be appreciated, as μ becomes larger,
the sparsity increases, but the performance becomes worse.