Sparse Command Generator for Remote Control
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Sparse Command Generator forRemote Control
Masaaki Nagahara (Kyoto Univ.)Daniel E. Quevedo (The Univ. of Newcastle)
Jan Østergaard (Aalborg Univ.)Takahiro Matsuda (Osaka Univ.)Kazunori Hayashi (Kyoto Univ.)
Remote Control System
RobotComamndGenerator
In remote control (RC), one has to transmit control commands through rate-limited networks such as wireless networks.
𝜃
Remote Control in Sparse Land
For rate-limited networks, control commands should be compressed.Sparse Representation can effectively compress control commands without much distortion.
RobotComamndGenerator
𝜃
Table of Contents
• Remote Control Systems– Energy-limiting control
• Sparsity-promoting method for RC– optimization– Fast algorithm (iterative-shrinkage algorithm)
• Examples• Conclusion
Table of Contents
• Remote Control Systems– Energy-limiting control
• Sparsity-promoting method for RC– optimization– Fast algorithm (iterative-shrinkage algorithm)
• Examples• Conclusion
Remote Control Systems
Given target points, find a control input such that the plant output fits the target points.
Radio Control Helicopter
�̇�=𝐴𝑥+𝐵𝑢𝑦=𝐶𝑥
𝑦𝑢Target points𝑃
𝑃𝑦
𝑢
Remote Control Systems
𝑌 1
𝑌 2
𝑌 𝑁
𝑡1𝑡 2 𝑡𝑁
min𝑢∈𝐿2
∑𝑖=1
𝑁
|𝑦 (𝑡𝑖 )−𝑌 𝑖|𝟐+𝜇∫0
𝑡𝑁
𝑢 (𝑡 )2𝑑𝑡
𝑃𝑦
𝑡�̇�=𝐴𝑥+𝐵𝑢𝑦=𝐶𝑥
𝑦𝑢
Tracking error on the sampling instants Energy limitation
Regularization parameter for tradeoff betweentracking error and control energy
Solution to Energy-limiting control
𝑢𝑜𝑝𝑡 (𝑡 )=∑𝑖=1
𝑁
𝜃 𝑖𝑔𝑖(𝑡)
𝜃= (𝜇 𝐼+𝐺𝑇𝐺 )− 1𝐺𝑌
[S. Sun et al., IEEE TAC, 2000]
𝑔𝑖 (𝑡 )=𝐶𝑒𝐴 (𝑡 𝑖−𝑡 ) 𝐵 , 𝑡∈ ¿0 , otherwise
𝐺= {(𝑔𝑖 ,𝑔 𝑗 )}𝑖=1 :𝑁 , 𝑗=1 :𝑁
The optimal control is given by
Remote Control System by Energy-limiting () Optimization
𝑢𝑜𝑝𝑡 (𝑡 )=∑𝑖=1
𝑁
𝜃 𝑖𝑔𝑖(𝑡)
(𝜇 𝐼+𝐺𝑇𝐺 )−1𝐺 𝑔 (𝑡) 𝑃𝑢 𝑦𝑌 𝜃
Reference vector
optimization(matrix multiplication)
Transmitted vector
D/A conversionActuator
Control input
Plant
Output
Table of Contents
• Remote Control Systems– Energy-limiting control
• Sparsity-promoting method for RC– optimization– Fast algorithm (iterative-shrinkage algorithm)
• Examples• Conclusion
• Energy-limiting optimization gives the optimal vector , the solution of -norm regularization:
• Sparsity-promoting optimization (-norm regularization, optimization):
Sparsity-Promoting Optimization
𝜃2∗=min
𝜃‖𝐺𝜃−𝑌‖2
2+𝜇‖𝜃‖22
𝜃1∗=min
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
Sparsity-Promoting Optimization
• -norm regularization produces a dense vector like
• -norm regularization (or optimization) produces a sparse vector like
• Sparse vectors can be compressed more effectively than a dense vector.– c.f. JPEG image compression producing sparse data in
the wavelet domain
𝜃2∗=[−2.6 ,−0.1 ,−1.8 ,0.1 ,−0.6 ]𝑇
𝜃1∗=[−2.6 ,0.09 ,−2.2 ,0 ,0 ]𝑇
Why does promote sparsity?
• By using the Lagrange dual, we obtain
for some .
{𝜃∈𝑅2:‖𝜃‖1=const }
0
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
¿argmin𝜃
‖𝜃‖1❑s . t .‖𝐺𝜃−𝑌‖2
2≤𝜖
{𝜃∈𝑅2:‖𝐺𝑌 −𝑌‖22≤𝜖 }
Feasible set
ball
-constrained optimization
Why does promote sparsity?
• By using the Lagrange dual, we obtain
for some .
{𝜃∈𝑅2:‖𝜃‖1=const }
0
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
¿argmin𝜃
‖𝜃‖1❑s . t .‖𝐺𝜃−𝑌‖2
2≤𝜖
{𝜃∈𝑅2:‖𝐺𝑌 −𝑌‖22≤𝜖 }
Feasible set
ball
𝜃1∗
Sparse!
Why does promote sparsity?
• By using the Lagrange dual, we obtain
for some .
{𝜃∈𝑅2:‖𝜃‖2=const }
0
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
¿argmin𝜃
‖𝜃‖1❑s . t .‖𝐺𝜃−𝑌‖2
2≤𝜖
{𝜃∈𝑅2:‖𝐺𝑌 −𝑌‖22≤𝜖 }
Feasible set
𝜃2∗
Not sparseball
How to solve Iterative-Shrinkage Algorithm
• The solution of
can be effectively obtained via a fast algorithm.𝜃 𝑗+1=𝑆2 𝜅 /𝑐( 1𝑐 𝐺𝑇 (𝑌 −𝐺𝜃 𝑗 )+𝜃 𝑗) , 𝑗=0,1,2 ,…
[Beck-Teboulle, SIAM J. Imag. Sci., 2009][Zibulevsky-Elad, IEEE SP Mag., 2010]
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
How to solve Iterative-Shrinkage Algorithm
• The solution of
can be effectively obtained via a fast algorithm.𝜃 𝑗+1=𝑆2 𝜅 /𝑐( 1𝑐 𝐺𝑇 (𝑌 −𝐺𝜃 𝑗 )+𝜃 𝑗) , 𝑗=0,1,2 ,…
[Beck-Teboulle, SIAM J. Imag. Sci., 2009][Zibulevsky-Elad, IEEE SP Mag., 2010]
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
𝑆2𝜅 / 𝑐 (𝑢)
𝑢2𝜅 /𝑐
−2𝜅 /𝑐 𝑐>𝜆max (𝐺𝑇𝐺)
Proposed Remote Control
Optimization 𝑔 (𝑡) 𝑃𝑢 𝑦𝑌 𝜃
𝜃1∗=argmin
𝜃‖𝐺𝜃−𝑌‖2
2+𝜅‖𝜃‖1❑
Fast Algorithm𝑢 (𝑡 )=∑
𝑖=1
𝑁
𝜃𝑖𝑔𝑖(𝑡)
A simple way to send a sparse vector
• Sparsify the reference via
• Send sparse vector • At the receiver, produce the control via
• This can be used when the transmitter is cheap and cannot accept an intelligent algorithm
𝑢 (𝑡 )=∑𝑖=1
𝑁
𝜃𝑖𝑔𝑖 (𝑡 ) ,𝜃=𝐺− 1𝜂
𝜂❑∗=argmin
𝜂‖𝜂−𝑌‖2
2+𝜆‖𝜂‖1❑=𝑆2𝜆(𝑌 )
𝑆2𝜆 (𝑌 ) 𝑔 (𝑡) 𝑃𝑢 𝑦𝑌 𝜃
𝐺− 1𝜂
Table of Contents
• Remote Control Systems– Energy-limiting control
• Sparsity-promoting method for RC– optimization– Fast algorithm (iterative-shrinkage algorithm)
• Examples• Conclusion
Examples
• Controlled plant:
• Reference data:
• Strategies:1: Energy-limiting design (regularization)2: Sparsity-promoting design ()3: Simple design (sparsifying via )
Vectors to be sent
Control input
Control input by the sparsity-promoting method has almostthe same energy (norm) as that by the energy-limiting method.The simple method does not limit the control size.
Plant output
Tracking error
The performances by and are almost the same.
Quantizing control vectors
We quantize the vectors by a uniform quantizer to encode them.
Tracking error with quantization
The -optimized control leads to large error due to quantization.
Conclusion• Sparsity-promoting optimization () for
remote control.• Sparse representation of leads to efficient compression
of transmitted signals.• Sparse vectors can be effectively obtained via a fast
algorithm.• Examples show the effectiveness of our method.
Thank you for your attention!