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!