M. Nagahara, D. E. Quevedo, T. Matsuda, and K. Hayashi, Compressive sampling for networked feedback control, IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), pp. 2733-2736, Mar., 2012.

1. 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 BSTRACT N ETWORKED C ONTROL S YSTEM 1 2
S OLUTION BY R ANDOM S AMPLING AND - O PTIMIZATION
We investigate the use of compressive sampling Problem Formulation However, if M is very large, it may take very
for networked feedback control systems. The method Rate-limited network
long time to compute the optimal vector. There-
proposed serves to compress the control vectors Given reference signal rk ∈ VM , k = 0, 1, . . . , find fore, we adopt random sampling as used in com-
Controller Decoder Plant
which are transmitted through rate-limited chan- the control uk ∈ VM (or the vector θ[k]) that mini- pressed sampling.
r(t) θ[k] u(t) y(t)
nels without much deterioration of control per-
formance. The control vectors are obtained by
K Ψ P mizes
T
The cost function J in (*) is then reduced to
1 2 2 2
an - optimization, which can be solved very Sensor J := |yk (t) − rk (t)| dt + µ θ[k] 0 (∗) J = Φθ[k] − α[k] 2 + µ θ[k] 0 ,
efficiently by FISTA (Fast Iterative Shrinkage- x[k] xc (t) 0
Thresholding Algorithm). Simulation results S where θ[k] is the Fourier coefficient vector:
where Φ ∈ RK×N (K < N ) and α[k] ∈ RK are
show that the proposed sparsity-promoting con- independent of θ[k].
trol scheme gives a better control performance than a M
conventional energy-limiting L2 -optimal control. Analog channel 1
- 2
Optimization
Digital channel uk (t) = θm [k]ψm (t)
m=−M
The optimization is executed in the feedback loop,
and hence the optimal vector should be computed
S PARSITY IN C ONTROL C ONTROL P ROBLEM Random Sampling as fast as possible, since a delay leads to instability
• The plant P is located away from the con- The control objectives: of the feedback system. Therefore, we relax the
To obtain the vector θ[k], one has to sample the cost function as
troller K. Remote control robots, aircrafts, continuous-time signal rk . Since rk is assumed to
and vehicles are examples. 1. Good tracking performance: r(t) ≈ y(t)
be band-limited up to ωM = 2πM/T [rad/sec], J = Φθ[k] − α[k] 2
2 + µ θ[k] 1 ,
2. Sparse control vector: θ[k] 0 N one can sample rk at a sampling frequency higher
1 2
than 2ωM , by Shannon’s sampling theorem. and solve this - optimization via FISTA.
Notation: Let T > 0 be the sampling period of
the control system. For a continuous-time signal
v on [0, ∞), we denote by vk , k = 0, 1, . . . , the
restriction of v to [kT, kT + T ), that is, S IMULATION R ESULTS
vk (t) := v(kT + t), t ∈ [0, T ), k = 0, 1, . . . Control vector θ[k] Control performance ek 2 and sparsity
Tracking error and sparsity
0
Assumption: Let M be a given positive inte- L1−L2 optimization
Error ||ek||2 [dB]
−5
10 10 Truncated L2 optimization
ger and define the signal subspace −10
• We should use a rate-limited network (e.g., −15
−20
wireless communication network, or the In- VM := span{ψ−M , . . . , ψM } −25
|θm[k]|
|θm[k]|
0 10 20 30 40 50 60 70 80 90 100
ternet) for sending the control signal (vector 1 1 1
ψm (t) := √ exp(jωm t), t ∈ [0, T ), 80
θ[k]) to the plant P . T
Sparsity ||θ[k]||0
60
ωm := 2πm/T. 40
• Sparse vectors can be encoded more effec- 20
tively than dense vectors obtained by usual We assume that rk ∈ VM and uk ∈ VM . That is, 0.1
−50 0 50
0.1
−50 0 50 0
m m 0 10 20 30 40 50 60 70 80 90 100
energy-limiting optimization as in Linear the reference signal r and the control signal u are
k
Quadratic (LQ) Optimal Control. For exam- band-limited up to the Nyquist frequency ωM = Left: proposed, Right: conventional The sparsity of the control vector obtained by the proposed
ple, we quantize the non-zero coefficients 2πM/T on each time interval [kT, kT + T ). method is about 8 out of 101 in the steady state.
in θ[k] and, in addition, send information
about the coefficient locations.
• In many cases, sparse vectors have small 1 C ONCLUSION R EFERENCES
norm. This leads to robustness against uncer- [A] M. Nagahara and D.E. Quevedo, IFAC
We have studied the use of compressive sam- fectively compress the signals transmitted. Future
World Congress, pp. 84–89, 2011.
tainty (or perturbation) in the plant model pling for feedback control systems with rate- work could include further investigation of bit- [B] M. Nagahara, T. Matsuda, and K. Hayashi,
(c.f. small gain theorem). limited communication channels. Simulation rate issues and the study of closed loop stability. IEICE Trans. Fundamentals, Vol. E95-A, No. 4,
studies indicate that the method proposed can ef- 2012.