C OMPRESSIVE S AMPLING FOR N ETWORKED F EEDBACK C ONTROL?

†‡

Masaaki Nagahara , Daniel E. Quevedo , Takahiro Matsuda , Kazunori Hayashi?

? Kyoto University, † The University of Newcastle, ‡ Osaka University

A BSTRACTN ETWORKED C ONTROL S YSTEM

We investigate the use of compressive samplingfor networked feedback control systems. The method
proposed serves to compress the control vectors
which are transmitted through rate-limited channels without much deterioration of control performance. The control vectors are obtained by1 2an ` -` optimization, which can be solved veryefficiently by FISTA (Fast Iterative ShrinkageThresholding Algorithm).
Simulation results
show that the proposed sparsity-promoting control scheme gives a better control performance than aconventional energy-limiting L2 -optimal control.

r(t)KDecoderθ[k]ΨPlant

u(t)

P

y(t)

Sensorx[k]Sxc (t)Given reference signal rk ∈ VM , k = 0, 1, . . . , findthe control uk ∈ VM (or the vector θ[k]) that minimizesZ T2J :=|yk (t) − rk (t)| dt + µkθ[k]k0(∗)
where θ[k] is the Fourier coefficient vector:

Analog channel
Digital channel

C ONTROL P ROBLEM• The plant P is located away from the controller K. Remote control robots, aircrafts,and vehicles are examples.

The control objectives:
1. Good tracking performance: r(t) ≈ y(t)2. Sparse control vector: kθ[k]k0  N
Notation: Let T > 0 be the sampling period of
the control system. For a continuous-time signalv on [0, ∞), we denote by vk , k = 0, 1, . . . , therestriction of v to [kT, kT + T ), that is,vk (t) := v(kT + t), t ∈ [0, T ), k = 0, 1, . . .

However, if M is very large, it may take very
long time to compute the optimal vector. Therefore, we adopt random sampling as used in compressed sampling.
The cost function J in (*) is then reduced toJ = kΦθ[k] − α[k]k22 + µkθ[k]k0 ,0uk (t) =M
X

where Φ ∈ RK×N (K < N ) and α[k] ∈ RK are
independent of θ[k].
`1 -`2 Optimization

θm [k]ψm (t)m=−M

S PARSITY IN C ONTROL

2S OLUTION BY R ANDOM S AMPLING AND ` -` O PTIMIZATIONProblem Formulation

Rate-limited network
Controller

1Random Sampling
To obtain the vector θ[k], one has to sample thecontinuous-time signal rk . Since rk is assumed tobe band-limited up to ωM = 2πM/T [rad/sec],
one can sample rk at a sampling frequency higher
than 2ωM , by Shannon’s sampling theorem.

The optimization is executed in the feedback loop,
and hence the optimal vector should be computed
as fast as possible, since a delay leads to instabilityof the feedback system. Therefore, we relax thecost function as
J = kΦθ[k] − α[k]k22 + µkθ[k]k1 ,and solve this `1 -`2 optimization via FISTA.

S IMULATION R ESULTS
Control vector θ[k]Control performance kek k2 and sparsity
Tracking error and sparsity

1• In many cases, sparse vectors have small `norm. This leads to robustness against uncertainty (or perturbation) in the plant model(c.f. small gain theorem).

1√ψm (t) := T exp(jωm t), t ∈ [0, T ),Error ||ek||2 [dB]10|θm[k]|VM := span{ψ−M , . . . , ψM }101−15−20
1020304050607080901001020304050k60708090100180

0.1
−500m

500.1
−500m

50

Left: proposed, Right: conventional

60402000The sparsity of the control vector obtained by the proposed
method is about 8 out of 101 in the steady state.

R EFERENCESC ONCLUSION
We have studied the use of compressive sampling for feedback control systems with ratelimited communication channels.
Simulation
studies indicate that the method proposed can ef-−10−250ωm := 2πm/T.
We assume that rk ∈ VM and uk ∈ VM . That is,
the reference signal r and the control signal u are
band-limited up to the Nyquist frequency ωM =2πM/T on each time interval [kT, kT + T ).

L1−L2 optimization
Truncated L2 optimization−5Sparsity ||θ[k]||0• Sparse vectors can be encoded more effectively than dense vectors obtained by usualenergy-limiting optimization as in Linear
Quadratic (LQ) Optimal Control. For example, we quantize the non-zero coefficientsin θ[k] and, in addition, send information
about the coefficient locations.

Assumption: Let M be a given positive integer and define the signal subspace|θm[k]|• We should use a rate-limited network (e.g.,
wireless communication network, or the Internet) for sending the control signal (vectorθ[k]) to the plant P .

0fectively compress the signals transmitted. Future
work could include further investigation of bitrate issues and the study of closed loop stability.

[A] M. Nagahara and D.E. Quevedo, IFAC
World Congress, pp. 84–89, 2011.
[B] M. Nagahara, T. Matsuda, and K. Hayashi,IEICE Trans. Fundamentals, Vol. E95-A, No. 4,2012.