Packetized Predictive Control for Rate-Limited Networks via Sparse Representation

Packetized Predictive Control for Rate-Limited Networksvia Sparse Representation

Masaaki Nagahara1 Daniel E. Quevedo2 Jan Østergaard3

1Kyoto University

2The University of Newcastle

3Aalborg University

2012 Dec. 10IEEE CDC 2012

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Motivation

Networked Control

Packet dropouts

→ Packetized Predictive Control

Bit-rate limitation

→ Sparse Representation

Plant

Controller

unreliable and rate-limited network

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Motivation

Networked Control

Packet dropouts → Packetized Predictive Control

Bit-rate limitation

→ Sparse Representation

Plant

Controller


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Motivation

Networked Control

Packet dropouts → Packetized Predictive Control

Bit-rate limitation → Sparse Representation

Plant

Controller


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Table of Contents

1 Packetized predictive control (PPC)

2 Sparsity optimization in PPC

3 Stability theorem

4 Optimization via Orthogonal Matching Pursuit (OMP)

5 Simulation results

6 Conclusion

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Table of Contents

1 Packetized predictive control (PPC)

2 Sparsity optimization in PPC

3 Stability theorem

4 Optimization via Orthogonal Matching Pursuit (OMP)

5 Simulation results

6 Conclusion

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Packetized Predictive Control

Suppose the plant is modeled by

x(k + 1) = Ax(k) +Bu(k), k = 0, 1, . . . , x(0) = x0 ∈ Rn.

Compute a tentative control sequence u0, u1, . . . , uN−1 for a finitehorizon (length N) of future time instants based on state observationx(k) and state prediction x0|k := x(k),x1|k, . . . ,xN−1|k.

Transmit the control sequence as a packet u = [u0, u1, . . . , uN−1]>

to a buffer.If a packet is dropped out, use a ”future” control ui (i ≥ 2)stemming from a previously received packet stored in the buffer.

PlantBufferController

u(x(k))x(k) u(k) x(k)

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x(0) u(x(0)) x(0)u(0)

At time k=0, compute the control packet u(x(0))using the observation x(0).

u(x(0))

u0 u2u1 u3

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x(0) u(x(0)) x(0)u(0)

The control packet u(x(0)) is successfully transmitted,and then the packet is stored in the buffer.

u(x(0))

u0 u2u1 u3

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x(0) u(x(0)) x(0)

Use the first element u0 in the buffer as the control input u(0).

u(x(0))

u0 u2u1 u3

u(0)

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At time k=1, compute the control packet u(x(1))using the observation x(1).

x(1) u(x(1)) x(1)

u0 u2u1 u3

u(x(1))

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The control packet u(x(1)) is transmitted to the buffer.

x(1) u(x(1)) x(1)

u0 u2u1 u3

u(x(1))

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The control packet u(x(1)) is transmitted to the buffer.

x(1) u(x(1)) x(1)

u0 u2u1 u3

u(x(1)) Packet dropout occurs!

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Use the second element of u(x(0)) stored in the bufferas the control u(1) at k=1.

x(1) u(x(1)) x(1)

u0 u2u1 u3

u(x(1)) u(x(0))

u0 u2u1 u3

u(1)

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Quadratic Packetized Predictive Control

In a standard PPC, we minimize the following quadratic (or `2) costfunction for the control packet u(x(k)), k = 0, 1, 2, . . . :

J(u) =∥∥xN |k

∥∥2P+

N∑i=0

∥∥xi|k∥∥2Q+ λ‖u‖22,

where u = [u0, . . . , uN−1]> ∈ RN and

x0|k = x(k), xi+1|k = Axi|k +Bui, i = 0, 1, . . . , N − 1.

A large horizon length N leads to robustness against packet dropouts,but it increases the size of the packet, which should be avoided forrate-limited networks.

Can we reduce the data size of the packet without reducing thehorizon length N?

Use sparse representation of the packet.

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J(u) =∥∥xN |k

∥∥2P+

N∑i=0

∥∥xi|k∥∥2Q+ λ‖u‖22,

where u = [u0, . . . , uN−1]> ∈ RN and





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J(u) =∥∥xN |k

∥∥2P+

N∑i=0

∥∥xi|k∥∥2Q+ λ‖u‖22,

where u = [u0, . . . , uN−1]> ∈ RN and





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J(u) =∥∥xN |k

∥∥2P+

N∑i=0

∥∥xi|k∥∥2Q+ λ‖u‖22,

where u = [u0, . . . , uN−1]> ∈ RN and





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Sparse Control Packet Design

Idea

Sparsify the control packet (vector) with the sparsity-promotingoptimization:

u(x(k)) , argminu∈RN

‖u‖0

subject to

‖xN |k‖2P +

N−1∑i=0

‖xi|k‖2Q ≤ x(k)>Wx(k).

Sparsity index ‖u‖0‖u‖0 is the number of nonzero elements in u = [u0, u1, . . . , uN−1]

>.

Trade-off parameter W

W is a positive semi-definite matrix specifying the trade-off between thesparsity and control performance.

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Sparse vectors can be effectively encoded by simple means.

Assume memoryless uniform scalar quantizer for encoding u.

u0 u1 u2 u3 u4 u5 u6 u7

û0 û1 û2 û3 û4 û5 û6 û7

8bit 8bit 8bit 8bit 8bit 8bit 8bit 8bit

Q

u0 0 0 u3 0 u5 u6 0

û0 û3 û5 û6

Q

8bit 8bit 8bit 8bit

8 ⇥ 4 = 32 bit

1 0 0 1 0 1 1 0 8 bit

+ location data

For dense vector

For sparse vector

40 bit

8 ⇥ 8 = 64 bit

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In general, assumeSampling frequency : fs > 0 [Hz]Horizon length: N ≥ 1Packet sparsity: S = ‖u‖0 < NQuantizer precision: b ≥ 1 [bit]For dense vectors (obtained by e.g., `2 optimization), one needs

N · b · fs [bit/sec]

For sparse vectors, quantizing the values and the location of thenonzero elements requires

S · b · fs︸︷︷︸for values

+ N · fs︸︷︷︸for location

= (Sb+N)fs [bit/sec]

If Nb > Sb+N , orS <

(1− b−1

)N

then sparse vectors can reduce the bit rate for transmission by((1− b−1)N − S

)bfs [bit/sec].

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Stability result

Assumption

The number of consecutive packet-dropouts is uniformly bounded by thehorizon length N . In other words, a packet will never be dropped outwhen the buffer is empty.



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Stability result

Theorem

For every Q > 0, there exist matrices P > 0 and W > 0 in theoptimization

u(x(k)) = argminu∈RN

‖u‖0, s.t. ‖xN |k‖2P +N−1∑i=0

‖xi|k‖2Q ≤ x(k)>Wx(k)

such that the networked control system is asymptotically stable, i.e.,limk→∞ x(k) = 0. The procedure to obtain such P and W is shown inthe article.



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How to solve it?

The optimization


‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0


can be rewritten as


‖u‖0, s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k)

for some matrices G and H.

The optimization is combinatorial, and hence hard to solve.

A greedy algorithm called Orthogonal Matching Pursuit (OMP) canbe used.

OMP may give a local minimum, but it always gives a feasiblesolution, and hence leads to asymptotic stability.

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How to solve it?

The optimization


‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0


can be rewritten as


‖u‖0, s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k)





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How to solve it?

The optimization


‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0


can be rewritten as


‖u‖0, s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k)





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Simulation Results

Controlled plant (unstable): a linearized model of an aircraft[Maciejowski, Predictive Control with Constraints]

xc = Acxc +Bcu,

Ac =

−1.2822 0 0.98 0

0 0 1 0−5.4293 0 −1.8366 0−128.2 128.2 0 0

, Bc =

−0.3

0−17

0

.poles: 0, 0,−1.5594± j2.2900Discrete-time model is obtained via zero-order-hold discretization withsampling time 0.5 (sec).

Horizon length (= packet size): N = 10

Packet-dropout probability: 50%if there have been N − 1 = 9 consecutive dropouts, we set the nextdropout probability to be 0.

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Simulation Results

Comparison:

1 OMP for the optimization (proposed)


‖u‖0 s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k)

2 `1-`2 optimization [Nagahara & Quevedo, IFAC, 2011], [Gallieri & Maciejovski, ACC, 2012]


(µ‖u‖1 +

1

2‖Gu−Hx(k)‖22

)3 `2 optimization (conventional)


(µ‖u‖22 +

1

2‖Gu−Hx(k)‖22

)

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Simulation Results

0 10 20 30 40 50 60 70 80 90 100

0

2

4

6

8

10

12

14

Sparsity ||u||0

k

||u||

0

OMP

Ideal

L2

L1/L2 (i)

L1/L2 (ii)

(OMP): argminu ‖u‖0 s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k).

(L1/L2): argminu(µ‖u‖1 + (1/2)‖Gu−Hx(k)‖22

)with (i) µ = 5.3× 103 and (ii) µ = 5.3

(L2): argminu(µ‖u‖22 + (1/2)‖Gu−Hx(k)‖22

)with µ = 3.1× 102 (reg) and µ = 0 (ideal).

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Simulation Results

0 10 20 30 40 50 60 70 80 90 10010

−40

10−20

100

[log plot] 2−norm of the state x(k)

k

log

10 ||x

(k)|

| 2

OMP

Ideal

L2

L1/L2 (i)

L1/L2 (ii)

0 10 20 30 40 50 60 70 80 90 100

0

1

2

3

4

5[linear plot] 2−norm of the state x(k)

k

||x(k

)|| 2

OMP

Ideal

L2

L1/L2 (i)

L1/L2 (ii)



)with (i) µ = 5.3× 103 and (ii) µ = 5.3



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Simulation Results

0 10 20 30 40 50 60 70 80 90 10010

−6

10−5

10−4

10−3

10−2

Computational time

k

Tim

e (

sec)

OMP

Ideal

L2

L1/L2 (i)

L1/L2 (ii)



)with (i) µ = 5.3× 103 and (ii) µ = 5.3



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Conclusion

We have proposed Packetized predictive control for packet dropoutswith sparse representation for rate-limited networks

The control system is asymptotically stable.

The optimization can be effectively solved via Orthogonal MatchingPursuit (OMP).

Simulation results show effectiveness of the proposed method.

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Conclusion

We have proposed Packetized predictive control for packet dropoutswith sparse representation for rate-limited networks

The control system is asymptotically stable.

The optimization can be effectively solved via Orthogonal MatchingPursuit (OMP).

Simulation results show effectiveness of the proposed method.

Mahalo!

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Packetized Predictive Control for Rate-Limited Networks via Sparse Representation

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