Sparse Representations for Packetized Predictive Networked Control

Masaaki Nagahara (Kyoto Univ.)Daniel E. Quevedo (The Univ. of Newcastle)

Networked Control in Sparse Land

• In networked control, one has to transmit control signals through unreliable networks.

• Packetized Predictive Control (PPC) can make the system robust against packet dropouts.

• Sparse Representation can effectively compress signals without much distortion.

• This work is

PPC + Sparse Representation

Table of Contents

• How does Packetized Predictive Control work?– PPC in networked control systems with packet

dropout• How can one obtain sparse vectors?

– -norm regularization– Fast iterative-shrinkage algorithm

• PPC + Sparse Representation– Is the feedback system stable? YES.

• Examples

Table of Contents

• How does Packetized Predictive Control work?– PPC in networked control systems with packet

dropout• How can one obtain sparse vectors?

– -norm regularization– Fast iterative-shrinkage algorithm

• PPC + Sparse Representation– Is the feedback system stable? YES.

• Examples

Packetized Predictive Control

• Compute a tentative control sequence for a finite horizon of future time instants.

• Transmit the sequence as a packet to a buffer.• If a packet is dropped out, use the control

stored in the buffer[Bemporad(1998), Casavola et al.(2006), Tang-Silva(2009), Quevedo(2007,2011)]

Controller Buffer Plant𝑥 (𝑘) 𝑈 (𝑘) 𝑢(𝑘) 𝑥 (𝑘)

Packetized Predictive Control

Controller Buffer Plant𝑥 (0) 𝑈 (0) 𝑥 (0)

𝑢0(0)𝑢1(0)

𝑢2(0)𝑢3(0)

𝑈 (0 )=[𝑢0 (0 ) ,𝑢1 (0 ) ,𝑢2 (0 ) ,𝑢3 (0 ) ]𝑇

𝐽 (𝑈 )=‖𝑥 (3|0 )‖𝑃2+∑𝑖=0

3

‖𝑥 (𝑖|0 )‖𝑄2+𝜆‖𝑈‖𝑅

2

‖𝑣‖𝑃2 ≜𝑣𝑇 𝑃𝑣

minimizing the cost function:

Packetized Predictive Control

Controller Buffer Plant

Assumption: The first packet is successfully transmitted to the buffer.

𝑥 (0) 𝑈 (0) 𝑥 (0)

𝑢0(0)𝑢1(0)

𝑢2(0)𝑢3(0)

Packetized Predictive Control

Controller Buffer Plant

Assumption: The first packet is successfully transmitted to the buffer. the 4 values are stored in the buffer.

𝑢0(0)𝑢1(0)

𝑢2(0)𝑢3(0)

𝑥 (0) 𝑈 (0) 𝑥 (0)

Packetized Predictive Control

Controller Buffer Plant

Assumption: The first packet is successfully transmitted to the buffer. the 4 values are stored in the buffer.

𝑢0(0)𝑢1(0)

𝑢2(0)𝑢3(0)

𝑢 (0 )=𝑢0 (0)𝑥 (0) 𝑈 (0) 𝑥 (0)

Packetized Predictive Control

Controller Buffer Plant

𝑢0(1)𝑢1(1)

𝑢2(1)𝑢3(1)

𝑥 (1) 𝑈 (1) 𝑥 (1)

𝑈 (1 )=[𝑢0 (1 ) ,𝑢1 (1 ) ,𝑢2 (1 ) ,𝑢3 (1 ) ]𝑇

𝐽 (𝑈 )=‖𝑥 (3|1 )‖𝑃2+∑𝑖=0

3

‖𝑥 (𝑖|1 )‖𝑄2+𝜆‖𝑈‖𝑅

2

Packetized Predictive Control

Controller Buffer Plant

𝑢0(1)𝑢1(1)

𝑢2(1)𝑢3(1)

𝑥 (1) 𝑈 (1) 𝑥 (1)

Packetized Predictive Control

Controller Buffer Plant

Packet-dropout occurs!𝑢0(1)𝑢1(1)

𝑢2(1)𝑢3(1)

𝑥 (1) 𝑈 (1) 𝑥 (1)

Packetized Predictive Control

Controller Buffer Plant

Use in the bufferas the control 𝑢0(0)

𝑢1(0)𝑢2(0)

𝑢3(0)

𝑢 (1 )=𝑢1(0)𝑥 (1) 𝑈 (1) 𝑥 (1)

Design of control packets• At each step, we solve the following optimization

for the packet :

• The solution is given by linear transformation of the state :

𝐽 (𝑈 )=‖𝑥 (𝑁|𝑘 )‖𝑃2+∑𝑖=0

𝑁

‖𝑥 (𝑖|𝑘 )‖𝑄2+𝜆‖𝑈‖2

2

¿‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜆‖𝑈‖2

2+‖𝑥 (𝑘)‖𝑄2

Table of Contents

• How does Packetized Predictive Control work?– PPC in networked control systems with packet

dropout• How can one obtain sparse vectors?

– -norm regularization– Fast iterative-shrinkage algorithm

• PPC + Sparse Representation– Is the feedback system stable? YES.

• Examples

Sparsity-Promoting Optimization

• Energy-limiting optimization (-norm regularization):

• Sparsity-promoting optimization (-norm regularization, optimization):

𝑈∗ (𝑘 )=min𝑈

‖𝐺𝑈−𝐻𝑥 (𝑘 )‖22+𝜆‖𝑈‖2

2

𝑈∗ (𝑘 )=min𝑈

‖𝐺𝑈−𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖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

𝑈∗=[−2.6 ,−0.1 ,−1.8 ,0.1 ,−0.6 ]𝑇

𝑈∗=[−2.6 ,0.09 ,−2.2 ,0 ,0 ]𝑇

Why does promote sparsity?

• By using the Lagrange dual, we obtain

for some .

{U∈𝑅2 :‖𝑈‖1=const }

0

𝑈∗ (𝑘 )=argmin𝑈

‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖1

❑

¿argmin𝑈

‖𝑈‖1❑s . t .‖𝐺𝑈−𝐻𝑥 (𝑘 )‖2

2≤𝜖

{U∈𝑅2 :‖𝐺𝑈−𝐻𝑥‖22≤𝜖 }

Feasible set

ball

Why does promote sparsity?

• By using the Lagrange dual, we obtain

for some .

𝑈∗ (𝑘 )=argmin𝑈

‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖1

❑

¿argmin𝑈

‖𝑈‖1❑s . t .‖𝐺𝑈−𝐻𝑥 (𝑘 )‖2

2≤𝜖

{U∈𝑅2 :‖𝑈‖1=const }

{U∈𝑅2 :‖𝐺𝑈−𝐻𝑥‖22≤𝜖 }𝑈∗

0Sparse!

Feasible set

ball

Comparison with energy-limiting optimization

{U∈𝑅2 :‖𝑈‖2=const }

{U∈𝑅2 :‖𝐺𝑈−𝐻𝑥‖22≤𝜖 }𝑈∗

0Not sparse

¿argmin𝑈

‖𝑈‖22s . t .‖𝐺𝑈−𝐻𝑥 (𝑘 )‖2

2≤𝜖

𝑈∗ (𝑘 )=argmin𝑈

‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖2

2

Feasible set

ball

Iterative-Shrinkage Algorithm

• The solution of

can be effectively obtained via a fast algorithm.

𝑈∗ (𝑘 )=argmin𝑈

‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖1

❑

𝑈 𝑗+1=𝑆2𝜇 /𝑐 ( 1𝑐 𝐺𝑇 (𝐻𝑥 (𝑘)−𝐺𝑈 𝑗 )+𝑈 𝑗) , 𝑗=0,1,2,…

[Beck-Teboulle, SIAM J. Imag. Sci., 2009][Zibulevsky-Elad, IEEE SP Mag., 2010]

Iterative-Shrinkage Algorithm

• The solution of

can be effectively obtained via a fast algorithm.𝑈 𝑗+1=𝑆2𝜇 /𝑐 ( 1𝑐 𝐺𝑇 (𝐻𝑥 (𝑘)−𝐺𝑈 𝑗 )+𝑈 𝑗) , 𝑗=0,1,2,…

𝑆2𝜇/ 𝑐 (𝑢)

𝑢2𝜇 /𝑐

−2𝜇 /𝑐 𝑐>𝜆max (𝐺𝑇𝐺)

𝑈∗ (𝑘 )=argmin𝑈

‖𝐺𝑈 −𝐻𝑥 (𝑘 )‖22+𝜇‖𝑈‖1

❑

[Beck-Teboulle, SIAM J. Imag. Sci., 2009][Zibulevsky-Elad, IEEE SP Mag., 2010]

Table of Contents

• How does Packetized Predictive Control work?– PPC in networked control systems with packet

dropout• How can one obtain sparse vectors?

– -norm regularization– Fast iterative-shrinkage algorithm

• PPC + Sparse Representation– Is the feedback system stable? YES.

• Examples

Stability Analysis• The controlled plant: • The control packet:

• If then • This implies that asymptotic stability will not be

achieved if is unstable even if there is no packet-dropout.

{𝑥∈ℝ 2:‖𝐺𝑇 𝐻𝑥‖∞≤2𝜇}

0𝑥 (𝑘)

Practical Stability• Assumption: The number of consecutive packet-

dropouts is always less than the prediction horizon (the size of the buffer)

[Theorem]Let and choose to satisfy

where Then for we have (practical stability)

where are constants.

𝐽 (𝑈 )=‖𝑥 (𝑁|𝑘 )‖𝑃2+∑𝑖=0

𝑁

‖𝑥 (𝑖|𝑘 )‖𝑄2+𝜇‖𝑈‖1

❑

Terminal condition

Table of Contents

• How does Packetized Predictive Control work?– PPC in networked control systems with packet

dropout• How can one obtain sparse vectors?

– -norm regularization– Fast iterative-shrinkage algorithm

• PPC + Sparse Representation– Is the feedback system stable? YES.

• Examples

Examples

• Controlled plant

the elements in and are generated by random sampling from . has 3 unstable eigenvalues.

• The horizon length is .• Two designs:

– Sparsity-promoting design ()– Energy-limiting design (regularization)

Transmitted Control Packets

Histogram of Quantized Transmitted Values

Proposed

Conventional

-norm of the state

Sparsity-promoting (proposed)

Energy-limiting (Conventional)

-norm of the state

Sparsity-promoting (proposed)

Energy-limiting (Conventional)

Proposed method leads to more effective compression than the conventional method without much distortion.

Parameter vs sparsity and performance

𝐽 (𝑈 )=‖𝑥 (𝑁|𝑘 )‖𝑃2+∑𝑖=0

𝑁

‖𝑥 (𝑖|𝑘 )‖𝑄2+𝜇‖𝑈‖1

❑

Sparsity≜5−‖𝑈‖0=5−1100

∑𝑘=1

100

‖𝑈 (𝑘 )‖0

Performance≜‖𝑥‖2

Conclusion• Sparsity-promoting optimization () for packetized

predictive control.• Sparse representation of packets leads to efficient

compression of transmitted signals.• The feedback system can be practically stable.• Examples show the effectiveness of our method.• Future work may include

– Bit-rate analysis of optimized control– Robustness against disturbances in the plant

Conclusion• Sparsity-promoting optimization () for packetized

predictive control.• Sparse representation of packets leads to efficient

compression of transmitted signals.• The feedback system can be practically stable.• Examples show the effectiveness of our method.• Future work may include

– Bit-rate analysis of optimized control– Robustness against disturbances in the plant

Grazie!