COMPRESSIVE SAMPLING FOR NETWORKED FEEDBACK CONTROL
Masaaki Nagahara, Daniel E. Quevedo† , Takahiro Matsuda‡, Kazunori Hayashi†Graduate School of Informatics, Kyoto University
School of Electrical Engineering & Computer Science, The University of Newcastle‡
Graduate School of Engineering, Osaka University
ABSTRACT

We investigate the use of compressive sampling for networked feedback control systems. The method proposed
serves to compress the control vectors which are transmitted
through rate-limited channels without much deterioration ofcontrol performance. The control vectors are obtained by an1 -2 optimization, which can be solved very efficiently by
FISTA (Fast Iterative Shrinkage-Thresholding Algorithm).
Simulation results show that the proposed sparsity-promoting
control scheme gives a better control performance than aconventional energy-limiting L 2 -optimal control.
Index Terms— compressive sampling, networked control, 1 -2 optimization
1. INTRODUCTION
The objective of this article is to design a controller in a networked control system [2] that produces sparse control vectors for effective compression before transmissions. Unfortunately, the calculation of optimal sparse vectors will, in general, require significant computational cost and may thereby
introduce delays, which are unacceptable for closed-loop operation. To overcome this issue, we subsample the problem toreduce its size and adopt a fast algorithm called FISTA (FastIterative Shrinkage-Thresholding Algorithm) [1].
Networked control systems are those in which the controlled plants are located away from the controllers, and thecommunication should be made through rate-limited communication channels such as wireless networks or the Internet[8]. In networked control systems, efficient signal compression or representation is essential to send control data throughrate-limited communication channels. For this purpose, we
propose an approach of sparse control signal representation
using the compressive sampling technique [4]. Our contributions in this paper are (1) a new strategy for networked feedback control systems based on compressive sampling, (2) aneffective data compression scheme of the control signals withsparse representation, (3) formulation of the design problemby 1 -2 optimization which can be efficiently solved.
The compressive sampling approach will open up a newvista in control theory. To the best of our knowledge, so

rk

K

θ[k]

x[k]

Ψ

S

u(t)

P

y(t)

xc (t)

Fig. 1. Networked control system. The dotted line indicates a
rate-limited communication channel.
far only a few studies have applied compressive sampling tocontrol: [3] proposes to use compressive sensing in feedback
control systems for perfect state estimation and [6] proposes
sparse representation of transmitted control packets for feedback control with packet dropouts. For remote control systems, [7, 5] also propose to use  1 -2 optimization (as in thispaper). However, [7, 5] consider only feed-forward control
systems.
2. CONTROL PROBLEM
Fig. 1 shows the networked control system which we consider
in this article. The system consists of a controlled plant P , asensor (or sampler) S, a decoder or a digital-to-analog (DA)converter Ψ, and a digital controller K. The definitions ofthese systems are given as follows:
Plant P : The controlled plant P is modeled by the following state-space representation:ẋc (t) = Axc (t) + bu(t),P :(1)y(t) = c xc (t), t ∈ [0, ∞),where xc (0) = 0, A ∈ Rν×ν , and b, c ∈ Rν×1 .
Sensor S: The sensor (or sampler) S converts the continuoustime state xc into a discrete-time signal x[k] :=xc (kT ), k = 0, 1, 2, . . . , where T > 0 is the sampling period.
Decoder Ψ: The decoder (or DA converter) Ψ converts a
vector valued signal θ[k] into a continuous-time signal

{uk (t)}t∈[0,T ) , k = 0, 1, 2, . . . viaΨ : θ[k] → uk :=Mθm [k]ψm ∈ L2 [0, T ), (2)m=−Mwhere M is a positive integer, θ m [k] is the m-th element of the vector θ[k], and12πm, t ∈ [0, T ).
ψm (t) := √ exp(jωm t), ωm :=TT(3)
We call the vector θ[k] a control vector. Note that thecontinuous-time signal u k is band-limited to ω M =2πM/T [rad/sec]. That is, u k belongs to the following subspace of L 2 [0, T ):VM := span{ψ−M , . . . , ψM } ⊂ L2 [0, T ].

(4)

The input u to the plant P is defined by u(t + kT ) =uk (t), t ∈ [0, T ), k = 0, 1, 2, . . . .
Controller K: The controller K uses a continuous-time reference signal rk ∈ VM , k = 0, 1, 2, . . . , and the sampled state x[k] to produce the control vector θ[k]. The
latter defines the input signal u k as per (2).
We assume that S, Ψ, and K are synchronized at t = kT ,k = 0, 1, 2, . . . . We also assume that we can transmit thecontrol vector θ[k] and the sampled state x[k] through communication channels without any delays nor packet dropouts.
In this article, we consider a situation where the size N =2M +1 of θ[k] is much larger than the size ν of the state x[k],and should be compressed because θ[k] needs to be transmitted through a rate-limited communication channel.
Under these assumptions, we then formulate our control
problem. Let y k be the continuous-time signal y on the interval [kT, (k + 1)T ), that is, yk (t) := y(t + kT ), t ∈ [0, T ),k = 0, 1, . . . . We design the controller K to achieve the following objectives:
1. The first objective is to attenuate the tracking error between the reference r k and the output y k on the interval[kT, (k + 1)T ), k = 0, 1, 2, . . . . The error is measured
by the L2 norm:
 T22|||yk − rk |||2 :=|yk (t) − rk (t)| dt.
0

2. The second objective is to reduce the data size of thecontrol vector θ[k] which defines the control u k via(2). For this objective, we adopt the so-called 0-normof uk defined by |||u k |||0 := θ[k]0 , the number of thenonzero elements in θ[k].
In general, there is a trade-off between tracking-error attenuation and data-size reduction. For example, the sparsest solution uk ≡ 0 leads to very large error, and the control u k which

minimizes only the first objective function may not be sparse.
To solve this problem, we adopt regularization. The problem
is formulated as follows.
Problem 1 Given reference signal r k ∈ VM , k = 0, 1, 2, . . . ,find the control u k ∈ VM (or the control vector θ[k]) which
minimizes
J(uk ) := |||yk − rk |||22 + µ|||uk |||0 ,(5)

where µ > 0 is the regularization parameter to reconcile thetrade-off between the tracking error and the sparsity.
3. COMPRESSIVE SAMPLING FOR SPARSECONTROL VECTORSThe objective function J(u k ) in (5) is defined on an infinitedimensional signal subspace V M defined in (4). We here relax
the objective function into a finite-dimensional convex  1 -2
optimization problem by using the technique of compressive
sampling.
Since the signals rk and uk are assumed to be bandlimited up to the frequency ω M = 2πM/T [rad/sec], we cansafely discretize the signals by sampling them at a sampling
frequency higher than 2ω M based on Shannon’s sampling
theorem [9]. However, if M is very large, it may take very
long time to compute the optimal vector. It follows that there
may exist a large delay in the feedback loop, which may leadto instability and control performance deterioration. Hence
it is preferable to use a more efficient method than Shannon’s sampling. For this purpose, we adopt the technique ofcompressive sampling [4] with random sampling, which can
reduce the computational load for the optimization.
Random sampling is modeled as follows: we first split theinterval [0, T ) with sampling points t n := (n − 1)/fM , n =1, 2, . . . , N = 2M + 1, where fM := 2M/T is the Nyquist
rate. Then we randomly choose K sampling points (K <N ) from {t1 , . . . , tN }. To model this, we define a randommatrix U := [ei(1) , ei(2) , . . . , ei(K) ] ∈ {0, 1}K×N , wherei(1), . . . , i(N ) are discrete random variables chosen from theuniform distribution on {1, 2, . . . , N } such that i(l) < i(l +1), l = 0, 1, . . . , N − 1, and {e1 , . . . , eN } is the standard
basis in RN , that is, en (n = 1, 2 . . . , N ) denotes a unit vector
whose n-th element is equal to one and the other elements areequal to zero. The random matrix U is re-chosen at every
sampling step k.
By using the random variables i(1), i(2), . . . , i(K), we
define the random sampling points by t i(l) := i(l)h, h :=T /(N − 1), l = 1, 2, . . . , K < N . Then we consider random
sampling of the output y k . The sampled output y k (tn ) withthe control signal u k ∈ VM defined in (2) is computed byyk (tn ) = c exp(tn A)x0 +

M

m=−M

θm [k] φn , ψm ,(6)

where ·, · is the inner product in L 2 [0, T ), and
t ∈ [0, tn ),c exp [(tn − t)A] b,
φn (t) :=0,t ∈ (tn , T ].

µ vs error and sparsity
1.6

y[k] := [yk (ti(1) ), . . . , yk (ti(K) )] ∈ RK .

(7)

The minimization of the cost function (7) is still difficult
to solve when M is large since the optimization is a combinatorial one. To reduce this, we adopt a convex relaxation byreplacing the  0 norm with the  1 norm:
J1 (θ[k]) := Φθ[k] − α[k]22 + µθ[k]1 .

(8)

The cost function J 1 (θ[k]) in (8) is convex in θ[k] and hence
the optimal value uniquely exists. To obtain the  1 -2 optimal
vector, we use an iterative algorithm called FISTA [1]. Thisalgorithm is very simple and fast; it can be effectively implemented in digital devices, which leads to a real-time computation in the feedback loop. For this algorithm, see [1].
4. SIMULATION RESULTS
In this section, we illustrate simulation results to show theeffectiveness of the compressive sampling technique in networked feedback control systems.
The matrices in the state-space representation (1) of thecontrolled plant P are taken as


 
 
0
1
0
−α
A=
, b=
, c=
,
−αβ −α − β11where α = 5 and β = 10. We assume the initial state x(0) =0. The control period T is set to be 2π. The number of thebasis functions {ψm }, or the size of the control vector θ[k] is
N = 2M + 1 = 101 (M = 50). We use the referencerk (t) = sin(10t) + cos(5t), k = 0, 1, 2, . . . .
The sparsity of the reference r k is given by |||rk |||0 = 8
N = 101. Therefore, r k is a sparse vector when it is represented by the basis functions {ψ m } defined in (3). That

40

30

1

25

0.8

20

0.6

15

0.4

10

0.2

Then by (6), we have y[k] = U Gθ[k] + U Hx[k], where
G is an N × N matrix defined by (G) ij = φi , ψj , i =1, . . . , N , j = −M, . . . , M , and H is an N ×ν matrix defined
by H := exp(t1 A )c, . . . , exp(tN A )c . Let r[k] :=[rk (t1 ), rk (t2 ), . . . , rk (tN )] ∈ RN and Φ := U G, α[k] =U (r[k] − Hx[k]). Then the tracking error at the random sampling points {ti(1) , ti(2) , . . . , ti(K) } is given by y[k] − r[k] =Φθ[k] − α[k]. It follows that the cost function (5) in Problem
1 is approximately described in a finite-dimensional one:

2

0 −5
10

Sparsity ||θ||0

RMS Error

1.2

Define the randomly sampled output vector

J0 (θ[k]) := Φθ[k] − α[k]22 + µθ[k]0 .

1(Left) RMS Error for L −L opt.
2
(Left) RMS Error for L opt.
35
(Right) Sparsity ||θ||0 for L1−L2 opt.

1.4

5
10

−4

10

−3

−2

µ

10

−1

10

00
10

Fig. 2. Regularization parameter versus RMS tracking error
and average sparsity
is, the reference rk is sparse with respect to the basis {ψm }.
The shortest sampling interval in random sampling is h =T /(N − 1) = 2π/100. We set the number of random sampling K = 33. The iteration steps in FISTA for minimizing
the 1 -2 optimization in (8) is 10. We run the simulation of
the feedback control for k = 0, 1, . . . , 100, that is, the lengthof simulation time is T f := T × 101 = 202π.
First, we compute the relation between the regularization
parameter µ in (8) and metrics for control performance to beachieved by the optimal control vector θ[k]. We use two metrics: RMS (Root Mean Square) of the tracking error e := y−r
and the average sparsity of control vector. {θ[k]} 100
k=0 . TheRMS is defined as:


100
1 Tf
1 
|e(t)|2 dt =|||yk − rk |||22 .
RMS(e) :=
Tf 0
Tf
k=0

100 θ[k]0
The average sparsity is defined as: θ 0 :=k=0 101 .
Fig. 2 shows the performance as a function of the parameter µ.
To compare the proposed method with a conventional one, we
consider the L 2 -optimal control which minimizes J 2 (uk ) :=|||yk − rk |||22 + µ2 |||uk |||22 . This cost function limits the energy(i.e., the L2 norm) of the control u k , which has been widelyused in control. The optimal control vector, say θ 2 [k], is givenby(9)θ 2 [k] = (µ2 I + G G)−1 G (r[k] − Hx[k]),where we assume the control u k is in the subspace VM definedin (4). The RMS error performance for the L 2 -optimal control
is also shown in Fig. 2. In this case, the feedback system becomes unstable for µ 2 < 0.0005. The sparsity of the optimal
control vector is θ 2 [k]0 = 101 for all k = 0, 1, . . . , 101.
That is, the L2 -optimal control does not produce any sparse
vectors at all.
Fig. 2 suggests that the optimal parameter is µ ≈ 0.002
for 1 -2 optimization, and µ 2 ≈ 0.0005 for L2 optimization. With these parameters, we simulate the feedback control. Fig. 3 (a) shows the absolute value of the nonzero elements in the 1 -2 optimal control vector θ[k] at k = 50.
We can see that the number of the nonzero elements is 8 outm|θ [k]|

10m|θ [k]|

10

same data size. This shows the effectiveness of our method.
In an additional simulation study, we considered a step
function for the reference, that is, r k (t) = r ∈ R, k =0, 1, 2, . . . . This signal is also sparse in the space V M andproduces a sparse control vector (we omit details due to space
limitations).

1

1

5. CONCLUSION
0.1
−50

0
m

0.1
−50

50

(a) Proposed

0
m

50

(b) Conventional

Fig. 3. The absolute values of the nonzero coefficients ofθ[k] based on compressive sampling with  1 -2 optimization(left)and θ 2 [k] based on L 2 optimal control (right) at k = 50.
The stars in the right figure are the 8 elements of the truncatedvector.

We have studied the use of compressive sampling for feedback control systems with rate-limited communication channels. Simulation studies indicate that the method proposed
can effectively compress the signals transmitted. Control vectors are obtained via an  1 -2 optimization, which is solvedby the FISTA algorithm. Future work could include further
investigation of bit-rate issues and the study of closed loop
stability.
6. REFERENCES

Tracking error and sparsity

k 2

Error ||e || [dB]0

L1−L2 optimization
Truncated L2 optimization

−5
−10
−15
−20
−25
0

10

20

30

40

50

60

70

80

90

100

[2] A. Bemporad, M. Heemels, and M. Johansson, Networked Control Systems, Springer, Oct. 2010.

Sparsity ||θ[k]||0

80
60
40
20
0
0

[1] A. Beck and M. Teboulle, “A fast iterative shrinkagethresholding algorithm for linear inverse problems,”
SIAM J. Imaging Sciences, vol. 2, no. 1, pp. 183–202,
Jan. 2009.

10

20

30

40

50
k

60

70

80

90

100

Fig. 4. Top figure: Tracking error e k 2 in dB (solid: proposed, dash: conventional), bottom figure: sparsity θ[k] 0
of 101 (the size of the vector), and hence the vector is very
sparse. Then, the L 2 norm of the tracking error e k := rk − ykon the k-th period |||e k |||2 (k = 0, 1, . . . , 100) is shown in
the top figure in Fig. 4. The sparsity history {θ[k] 0 }100k=0
is shown in the bottom figure in Fig. 4.
To compare the proposed method with conventional L 2 optimal control, we compute the control vector θ 2 [k] by theformula (9) with µ 2 = 0.0005. Fig. 3 (b) shows the absolutevalue of the nonzero elements in the control vector θ 2 [k] at
k = 50. We can see that all the elements in this vector are
nonzero (cf Fig. 3 (a)). One may think that the vector θ 2 [k] iscompressible since almost all the elements are nearly zero. To
see the difference, we truncate the full vector θ 2 [k] by using
the sparsity history in Fig. 4.
The stars in Fig. 3 (b) are the 8 elements of the truncatedvector. The tracking error by the truncated vectors is shown in
the top figure in Fig.4. The proposed method shows the better
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