Sparse Command Generator for Remote Control
Masaaki Nagahara, Daniel E. Quevedo, Jan Østergaard, Takahiro Matsuda, and Kazunori Hayashi

Abstract— In this article, we consider remote-controlled systems, where the command generator and the controlled object
are connected with a bandwidth-limited communication link.
In the remote-controlled systems, efficient representation ofcontrol commands is one of the crucial issues because of the
bandwidth limitations of the link. We propose a new representation method for control commands based on compressedsensing. In the proposed method, compressed sensing reduces
the number of bits in each control signal by representing it as asparse vector. The compressed sensing problem is solved by an1 -2 optimization, which can be effectively implemented withan iterative shrinkage algorithm. A design example also showsthe effectiveness of the proposed method.

I. I NTRODUCTION
Compressed sensing has recently been a focus of intensive
researches in the signal processing community. It aims atreconstructing a signal by assuming that the original signal
is sparse [2]. The core idea used in this area is to introduce
a sparsity index in the optimization. The sparsity index of avector v is defined by the amount of nonzero elements in v
and is usually denoted by v 0 , called the “0 norm.” Thecompressed sensing decoding problem is then formulated
by least squares with  0 -norm regularization. The associatedoptimization problem is however hard to solve, since it is
a combinatorial one. Thus, it is common to introduce a
convex relaxation by replacing the  0 norm with the  1 norm[3]. Under some assumptions, the solution of this relaxed
optimization is known to be exactly the same as that of the0 -norm regularization [8], [2]. That is, by minimizing the
1 -regularized least squares, or by  1 -2 optimization, one
can obtain a sparse solution. Moreover, recent studies haveexamined fast algorithms for  1 -2 optimization [5], [1], [15].
The purpose of this paper is to investigate the use ofsparsity-inducing techniques for remote control [11], see [10]for an alternative approach. In remote-controlled systems,
control information is transmitted through bandwidth-limited
channels such as wireless channels [14] or the Internet [9].
M. Nagahara and K. Hayashi are with Graduate School of Informatics, Kyoto University, Kyoto 606-8501, Japan, nagahara@ieee.org,
kazunori@i.kyoto-u.ac.jp.
D. E. Quevedo is with School of Electrical Engineering & Computer
Science, The University of Newcastle, Callaghan, NSW 2308, Australia,
dquevedo@ieee.org.
J. Østergaard is with Department of Electronic Systems, Multimedia
Information and Signal Processing (MISP), Aalborg University, Niels Jernes
Vej 12, A6-305, DK-9220, Aalborg. jo@es.aau.dk
T. Matsuda is with Graduate School of Engineering, Osaka University,
Osaka, 5650871, Japan, matsuda@comm.eng.osaka-u.ac.jp
This research was supported in part by a Grand-in-Aid for Young
Scientists (B) of the Ministry of Education, Culture, Sports, Science andTechnology (MEXT) under Grant No. 22700069, No. 22760317, andNo. 21760289, and an Australian Research Council Australian Research
Fellowship (project number DP0988601).

There are two approaches to reduce the number of bits
transmitted on a wireless link, source coding and channel
coding approaches [4]. In the former, information compression techniques reduce the number of bits to be transmitted.
In the latter, efficient forward error-correcting codes reduce
redundant data (i.e., parity) in channel-coded information.
In this paper, we study the former approach and propose asparsity-inducing technique to produce sparse representationof control commands, which can reduce the number of bits
in transmitted data.
Our optimization to obtain sparse representation of control
commands is formulated as follows: we measure the tracking
error in the output trajectory of a controlled system byits 2 norm, and add an  1 penalty to achieve sparsity oftransmitted vector. This is an  1 -regularized  2 -optimization,or shortly  1 -2 -optimization, which is effectively solvedby the iterative shrinkage method mentioned above. The
problem of command generator has been solved when the
penalty is taken solely as an  2 norm, the solution of which
is given by a linear combination of base functions, called
control theoretic splines [13]. In this work, we also present
a simple method for achieving sparse control vectors when
the control commands are assumed to be in a subspace of
these splines. An example illustrates the effectiveness of our
method compared with the  2 optimization.
Notationn, the 1 and 2 norms
For a vector v = [v1 , . . . , vn ] ∈ Rnare
√ respectively defined by v 1 := i=1 |vi | and v2 :=v  v. For a real number x ∈ R,

1, if x ≥ 0,
sgn(x) :=, (x)+ := max{x, 0}.
−1, if x < 0,

We denote the determinant of a square matrix M by det(M ),and the maximum eigenvalue of a symmetric matrix M
by λmax (M ). Let L2 [0, T ] be the set of Lebesgue square
integrable functions on [0, T ]. For f, g ∈ L 2 [0, T ], the innerproduct is defined by T
f, g :=f (t)g(t)dt.
0

II. C OMMAND G ENERATION P ROBLEM
Let us consider the following linear SISO (Single-Input
Single-Output) plant:ẋ(t) = Ax(t) + bu(t),P :(1)y(t) = c x(t), t ∈ [0, ∞), x(0) = 0,

where A ∈ Rn×n , b ∈ Rn and c ∈ Rn . We assume thatthe system P is stable and the state space realization (1)is reachable and observable. The output reference signal isgiven by data points D := {(t 1 , Y1 ), (t2 , Y1 ), . . . (tN , YN )},where ti ’s are time instants such that 0 < t 1 < t2 < · · · <tN =: T . Our objective here is to design the control signalu(t) such that the output trajectory y(t) is close to the data
points Y1 ,. . . ,YN at t = t1 , . . . , tN , that is, y(ti ) ≈ Yi , i =1, . . . , N . To measure the difference between {y(t i )}N
i=1 and
,
we
adopt
the
square-error
costfunction{Yi }Ni=1E2 (u) =Ngi (t)
P (t) = c eAt b
0

Fig. 1. Control theoretic spline gi (t) (solid) and the impulse response
P (t) of the plant P (dots).

y ref(y(ti ) − Yi )2 ,ti

−1(µI + G)

θ ∗2

g(t)

u∗2

P

y

i=1

where we have made the dependence of y(t i ) on u ={u(t)}t∈[0,T ] through the system equation (1).
In principle, one can achieve perfect tracking, that is,
E2 = 0, by some input signal 1 . However, the optimal input
for perfect tracking has very large gain especially when
the number N is very large, and may lead to oscillation
between the sampling instants t 1 , . . . , tN . This phenomenon
is known as overfitting [12]. To avoid this, one can adopt
a regularization or smoothing technique. This method is to
add a regularization term Ω(u) to the cost function E 2 (u).
We formulate our problem as follows:
Problem 1: Given data D, find a control signal u which
minimizes the regularized cost function J(u) = E 2 (u) +µΩ(u), where µ > 0 is the regularization parameter which
specifies the tradeoff between minimization E 2 (u) and thesmoothness by Ω(u).
A well-known regularization is to use L 2 function forΩ(u), called the control theoretic smoothing spline [13], [6].
We review this in the next section.
III. 2 C OMMAND D ESIGN BY C ONTROL T HEORETICS MOOTHING S PLINES
For the problem given in section II, the following L 2 regularized cost function was considered in [13]: Tu(t)2 dt. (2)J2 (u) := E2 (u) + µΩ2 (u), Ω2 (u) :=0

The optimal control u ∗2 which minimizes J2 (u) is given by alinear combination of the following functions called control
theoretic splines [13], [6]:c eA(ti −t) b,
if ti > t,gi (t) :=(3)0,
if ti ≤ t,
see Fig. 1. More precisely, the optimal control for (2) is givenbyu∗2 (t) =Nθi gi (t) = g(t) θ∗2 ,

(4)

i=1

θ∗2 := (µI + G)−1 y ref ,(5)

1 The explicit form of this input is given by (4) and (5) in Section III,with µ = 0.

Fig. 2. Remote-controlled system optimized with J2 (u) in (2). The vectorθ ∗2 is transmitted through a communication channel.

where g(t) := [g1 (t), . . . , gN (t)] , y ref := [Y1 , . . . , YN ] ,and G is the Grammian matrix of {g 1 , . . . , gN }, defined by[G]ij := gi , gj , i, j = 1, . . . , N .
IV. 1 -2 C OMMAND D ESIGN FOR S PARSE R EMOTE
C ONTROL
In remote-controlled systems, we transmit the control
input u = {u(t)}t∈[0,T ] to the system P through a communication channel. Since {u(t)} t∈[0,T ] is a continuous-time
signal, we should discretize it.
An easy way to communicate information on the input
signal is to transmit the data y ref itself, and produce the inputu(t) by the formulae (4) and (5) at the receiver side. The
vector y ref is just an N -dimensional one, and much easier totransmit than the infinite-dimensional vector {u(t)} t∈[0,T ] .
An alternative method consists in transmitting the coefficient vector θ ∗2 given in (5) instead of the continuous-time
signal u. This procedure is shown in Fig. 2. In this procedure,
we fix the sampling instants t 1 , . . . , tN and the vector y ref
is given. We first compute the parameter vector θ ∗2 by (5),and transmit this through a communication channel. The
transmitted vector is received at the receiver, and then thecontrol signal u ∗2 (t) is computed by (4), and applied to theplant P . We assume that the time instants t 1 , . . . , tN are
shared at the transmitter and the receiver.
A problem of the above-mentioned strategies is that thecommunication channel is band-limited and therefore thevector to be transmitted has to be first quantized and encoded.
To solve this, we will seek a sparse representation of
the transmitted vector θ in accordance with the notion ofcompressed sensing [2], [7].
Define a subspace V of L 2 [0, T ] by


M



θj φj , θi ∈ R , (6)V := u ∈ L2 [0, T ] : u =
j=1

where φ1 , . . . , φM are linearly independent vectors in
L2 [0, T ]. Note that if M = N and φi = gi , i = 1, . . . , N
defined in (3), the optimal control u ∗2 (t) in (4) belongs tothis subspace2 . We assume that the control u is in V, thatis, we find a control u in this subset. Under this assumption,
the squared-error cost function E 2 (u) is represented by
E2 (u) =

N

i=1

2

(y(ti ) − Yi )2 = Φθ − y ref 2 ,

−κ/c

(7)

θ
0

κ/c

where [Φ]ij = gi , φj , i = 1, . . . , N , j = 1, . . . , M . To
induce sparsity in θ, we adopt  1 penalty on θ and introduce
the following mixed  1 -2 cost function:
J1 (θ) :=12Φθ − y ref 2 + κθ1 .
2(8)
Fig. 3.

Note that if φj 1 = 1 for j = 1, . . . , M , then the costfunction (8) is an upper bound of the following L 1 -L2 costfunction:J1 (u) =1E2 (u) + κΩ1 (u),2

Ω1 (u) = T
0|u(t)|dt.

As mentioned in the introduction, the  1 -regularized leastsquares optimization is a good approximation to one regularized by the  0 norm which counts the nonzero elementsin θ. Although the solution which minimizes J 1 (θ) cannot
be represented analytically as in (4), we can compute an
approximated solution by using a fast numerical algorithm.
The algorithm is described in the next section. By using thissolution, say θ ∗sparse , the optimal control u ∗1 can be obtainedfromu∗1 (t) =Nθi∗ φi (t) = φ(t) θ ∗sparse ,t ∈ [0, T ].

i=1

V. S PARSE R EPRESENTATION BY 1 -2 O PTIMIZATION
We here describe a fast algorithm for obtaining the optimal
vector θ ∗sparse . First, we consider a general case of optimization. Next, we simplify the design procedure in a special
case.
A. General case
The cost function (8) is convex in θ and hence theoptimal value θ ∗sparse uniquely exists. However, an analyticalexpression as in (5) for this optimal vector is unknown except
when the matrix Φ is unitary. To obtain the optimal vectorθ∗sparse , one can use an iteration method. Recently, a very fastalgorithm for the optimal  1 -2 solution has been proposed,
which is called iterative shrinkage [1], [15].
This algorithm is given by the following: Give an initialvalue θ[0] ∈ RM , and let β[1] = 1, θ  [1] = θ[0]. Fix aconstant c such that c > Φ 2 := λmax (Φ Φ). Execute the2 The functions {g , . . . , g } are linearly independent [13].
1
N

y ref

Nonlinear function sgn(θ)(|θ| − κ/c)+

FISTAθ ∗sparse

g(t)

u∗1

P

y

Fig. 4.
Remote-controlled system optimized with J1 (θ) in (8). The
vector θ∗sparse minimizing (8) is computed by the FISTA (Fast Iterative
Shrinkage-Thresholding Algorithm) given in (9), and transmitted througha communication channel.

following iteration 3 :1 Φ (y ref − Φθ [j]) + θ [j] ,c
1 + 1 + 4β[j]2,
β[j + 1] =2
β[j] − 1θ  [j + 1] = θ[j] +(θ[j] − θ[j − 1]),β[j + 1]
j = 1, 2, . . . ,θ[j] = Sκ/c(9)where the function S κ/c is defined for θ = [θ1 , . . . , θM ]
by
sgn(θ1 )(|θ1 | − κ/c)+..
Sκ/c (θ) := .
.
sgn(θM )(|θM | − κ/c)+
The nonlinear function sgn(θ)(|θ| − κ/c) in S κ/c is shownin Fig. 3. If c > Φ 2 , the above algorithm converges to the
optimal solution minimizing the  1 -2 cost function (8) for
any initial value θ[0] ∈ R M with a worst-case convergence
rate O(1/j 2 ) [5], [1]. The above algorithm is very simple
and fast; it can be effectively implemented in digital devices,which leads to a real-time computation of a sparse vectorθ∗sparse .
B. The case Φ = G
We here assume M = N and φi = gi , i = 1, 2, . . . , N ,that is, Φ = G. Since g1 , . . . , gN are linearly independent
3 Several methods have been proposed for the iterative shrinkage [15]. The
algorithm given here is called FISTA (Fast Iterative Shrinkage-Thresholding
Algorithm) [1].

y refη ∗sparse

Sν

Φ−1

g(t)

u∗1

P

y
tbp
TABLE I

Fig. 5. Remote-controlled system optimized with J(η) in (10). The vector
η is transmitted through a communication channel.

D ESIGNED VECTORSθ∗2
9.79942.79951.6544
1.66951.03580.0059-1.0231-1.7456-2.0234-2.2424-2.41535.18132vectors in L [0, T ], the Grammian matrix Φ = G is nonsingular. Let the control input u beu(t) =

N

θi gi (t) = g(t) θ,

i=1

and let η := Φθ. Then, by (7) we haveN(y(ti ) − Yi )2 = η − y ref 22 .

θ∗sparse
9.67274.562602.9973000-2.8678-0.6316-4.8575
04.4185η ∗sparse
0.45000.81600.95000.81600.45000-0.4500-0.8160-0.9500-0.8160-0.4500
0y ref
0.50000.86601.00000.86600.50000.0000-0.5000-0.8660-1.0000-0.8660-0.5000-0.0000

i=1

Consider the following  1 -2 cost function:1
J(η) = νη1 + η − y ref 22 .
(10)
2
The optimal solution η ∗sparse minimizing this cost function is
given analytically by

y
L1
L1−L2
Simple L1−L2
Original Curve1
0.8η∗sparse = Sν (y ref ).

(11)η ∗sparse ,Then we transmit this optimal vector
and at thereceiver we reconstruct the optimal control by u ∗1 (t) =g(t) Φ−1 η ∗sparse . Fig. 5 shows the remote-controlled systemwith the optimizer η ∗sparse . In this case, we compute (11) only
one time, while in the general case considered in Section V-A
we should execute the iteration algorithm (9).

0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1

VI. E XAMPLE
We here show an example of the sparse command generator. The state-space matrices of the controlled plant P isassumed to be


 


0
1
0
A=
, B=
, C= 1 0 .
−1 −2
1Note that the transfer function of the plant P is 1/(s + 1) 2 .
The sampling instants are given by t i = i × π/6, i =1, 2, . . . , 12, and the data Y 1 , . . . , Y12 is given by Yi = sin ti ,that is, we try to track the sine function y(t) = sin t in
one period [0, 2π]. We assume the base functions φ i in thesubspace V in (6) are the same as g i ’s, that is, we consider
the case Φ = G discussed in Section V-B. We design threesignals to be transmitted: the  2 -optimized vector θ ∗2 in(5), the sparse vector θ ∗sparse given in subsection V-A, andthe sparse vector η ∗sparse in (11). We set the regularization
parameters µ = 0.01, κ = 0.001, and ν = 0.05, seeequations (2), (8) and (10).
The obtained vectors are shown in Table I. We can seethat the vector θ ∗sparse is the sparsest due to the sparsityinducing approach. The second sparsest vector is η ∗sparse
which converts small elements in y ref to 0. The vector θ ∗2 is
not sparse.
Fig. 6 shows the plant outputs obtained by the above
vectors. The transient responses show relatively large errors

0

1

2

3

4

5

6

Fig. 6. The original curve (dots) and outputs: by 2 -optimal θ∗2 (dash),1 -2 -optimal θ∗sparse (solid), and simple 1 -2 -optimal η∗sparse (dash-dots).

error
0.14

L2
L1−L2
Simple L1−L2

0.12
0.1
0.08
0.06
0.04
0.02
0
0

1

2

3

4

5

6

Fig. 7. The reconstruction errors: by 2 -optimal θ∗2 (dash), 1 -2 -optimalθ ∗sparse (solid), and simple 1 -2 -optimal η∗sparse (dash-dots).

tbp

advantages of sparse representation in view of information
theory.

TABLE IIQ UANTIZED VECTORS
Q(θ ∗2 )
9.8
2.8
1.7
1.7
1.0
0.0
-1.0
-1.7
-2.0
-2.2
-2.4
5.2

Q(θ∗sparse )
9.7
4.6
0.0
3.0
0.0
0.0
0.0
-2.9
-0.6
-4.9
0.0
4.4

Q(η ∗sparse )
0.5
0.8
1.0
0.8
0.5
0.0
-0.5
-0.8
-1.0
-0.8
-0.5
0.0

R EFERENCES
Q(y ref )
0.5
0.9
1.0
0.9
0.5
0.0
-0.5
-0.9
-1.0
-0.9
-0.5
0.0

error with quantization
0.14

L2
L1−L2
Simple L1−L2

0.12
0.1
0.08
0.06
0.04
0.02
0
0

1

2

3

4

5

6

∗

Fig. 8. The reconstruction errors with quantization: by 2 -optimal θ̂ 2∗(dash), 1 -2 -optimal θ̂sparse (solid), and simple 1 -2 -optimal η̂ ∗sparse (dashdots).

because of the phase delay in the plant P (s) = 1/(s + 1) 2 .
Despite of sparsity in θ ∗sparse and η ∗sparse , the performances of
the reconstructed signals are comparable to that of the  2 optimal reconstruction by θ ∗2 . To see the difference between
these performances more precisely, we draw the reconstruction errors in Fig. 7. We can see that the errors by θ ∗2 andθ∗sparse are almost comparable, and the error by η ∗sparse isrelatively large.
Then we consider quantization. We use the uniform quantizer with step size 0.1 and simulate the output reconstruction. Table II shows the quantized vectors. Fig. 8 shows
the reconstruction error under quantization. The errors by
the sparse vectors θ ∗sparse and η ∗sparse still remains small
while the 2 -optimal reconstruction shows errors affected byquantization. This is because the zero-valued elements in thesparse vectors do not suffer from any quantization distortion.
VII. C ONCLUSION
In this paper, we have proposed to use sparse representation for command generation in remote control by  1 -2
optimization. An example illustrates the effectiveness of the
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