View
175
Download
0
Category
Tags:
Preview:
DESCRIPTION
Presented at IEEE CDC 2012, Maui, USA. http://control.disp.uniroma2.it/cdc2012/
Citation preview
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
1 / 33
Motivation
Networked Control
Packet dropouts
→ Packetized Predictive Control
Bit-rate limitation
→ Sparse Representation
Plant
Controller
unreliable and rate-limited network
2 / 33
Motivation
Networked Control
Packet dropouts → Packetized Predictive Control
Bit-rate limitation
→ Sparse Representation
Plant
Controller
unreliable and rate-limited network
3 / 33
Motivation
Networked Control
Packet dropouts → Packetized Predictive Control
Bit-rate limitation → Sparse Representation
Plant
Controller
unreliable and rate-limited network
4 / 33
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
5 / 33
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
6 / 33
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)
7 / 33
Packetized Predictive Control
PlantBufferController
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
8 / 33
Packetized Predictive Control
PlantBufferController
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
9 / 33
Packetized Predictive Control
PlantBufferController
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)
10 / 33
Packetized Predictive Control
PlantBufferController
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))
11 / 33
Packetized Predictive Control
PlantBufferController
The control packet u(x(1)) is transmitted to the buffer.
x(1) u(x(1)) x(1)
u0 u2u1 u3
u(x(1))
12 / 33
Packetized Predictive Control
PlantBufferController
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!
13 / 33
Packetized Predictive Control
PlantBufferController
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)
14 / 33
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.
15 / 33
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.
16 / 33
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.
17 / 33
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.
18 / 33
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.
19 / 33
Sparse Control Packet Design
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
20 / 33
Sparse Control Packet Design
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].
21 / 33
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.
PlantBufferController
u(x(k))x(k) u(k) x(k)
22 / 33
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.
PlantBufferController
u(x(k))x(k) u(k) x(k)
23 / 33
How to solve it?
The optimization
u(x(k)) , argminu∈RN
‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0
‖xi|k‖2Q ≤ x(k)>Wx(k)
can be rewritten as
u(x(k)) = argminu∈RN
‖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.
24 / 33
How to solve it?
The optimization
u(x(k)) , argminu∈RN
‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0
‖xi|k‖2Q ≤ x(k)>Wx(k)
can be rewritten as
u(x(k)) = argminu∈RN
‖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.
25 / 33
How to solve it?
The optimization
u(x(k)) , argminu∈RN
‖u‖0, s.t. ‖xN |k‖2P+N−1∑i=0
‖xi|k‖2Q ≤ x(k)>Wx(k)
can be rewritten as
u(x(k)) = argminu∈RN
‖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.
26 / 33
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.
27 / 33
Simulation Results
Comparison:
1 OMP for the optimization (proposed)
u(x(k)) = argminu∈RN
‖u‖0 s.t. ‖Gu−Hx(k)‖22 ≤ x(k)>Wx(k)
2 `1-`2 optimization [Nagahara & Quevedo, IFAC, 2011], [Gallieri & Maciejovski, ACC, 2012]
u(x(k)) = argminu∈RN
(µ‖u‖1 +
1
2‖Gu−Hx(k)‖22
)3 `2 optimization (conventional)
u(x(k)) = argminu∈RN
(µ‖u‖22 +
1
2‖Gu−Hx(k)‖22
)
28 / 33
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).
29 / 33
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)
(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).
30 / 33
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)
(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).
31 / 33
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.
32 / 33
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!
33 / 33
Recommended