In Celebration of Bill Heltons 56 th Birthday NSF Workshop in Honor - - PowerPoint PPT Presentation

In Celebration of Bill Helton’s 56th Birthday

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 1 / 17

Performance of Networked Feedback Systems: Best Tracking and Optimal Power Allocation

Jie Chen

Department of Electronic Engineering City University of Hong Kong Hong Kong, China (On leave from University of California, Riverside)

In Honor of Professor Bill Helton San Diego, CA October 2-4, 2010

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 2 / 17

Background

Networked Control Systems

Controller Plant Channel r y

New issues and problems: packet loss, time delay, limited capacity, etc Channel models: quantized channel, AWGN channel, etc Degrading effects

• 1. Feedback stabilization
• 2. Control performance

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 3 / 17

Background

AWN Feedback Channel

K(s) P(s) + n(t)

The AWN channel has a prescribed power constraint, which has a ready connection to channel capacity. For minimum phase systems, the channel signal-to-noise ratio (SNR) must satisfy the bound [Braslavsky 2004] SNR > 2

• pi

for feedback stabilization.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 4 / 17

Tracking Performance Over an Additive White Noise Channel

Tracking Over an AWN Channel

Objective: Find how the tracking performance may be constrained by the channel Configuration

[K1 K2] P + n(t) r(t) y(t)

Parallel channel

+ + n1 u1 v1 nm um vm

Assumptions

• 1. The plant is MIMO, right-invertible and minimum phase.
• 2. The noise and the reference input are uncorrelated.
• 3. Parallel AWN channel: n(t) is white and

E[n(t)n(t)T] = diag(Φ1, . . . , Φm).

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 5 / 17

Tracking Performance Over an Additive White Noise Channel

Problem Statement

Tracking performance: E[e2] = E[r − y2]. The reference signal: r(t) = (r1(t), . . . , rm(t))T is a WSS real vector valued random process.

• Power of ri(t):

σ2

ri = E[ri(t)2].

• Power spectrum of r(t): positive real rational matrix

Gr(jω) = ψr(jω)ψT

r (−jω).

The channel has the input power constraint E[y2] ≤ Γ. Optimal tracking problem inf

K stabilizing P E[e2], subject to E[y2] ≤ Γ

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 6 / 17

Tracking Performance Over an Additive White Noise Channel

Approach

inf

K stabilizing P E[e2], subject to E[y2] ≤ Γ

• 1. Define

H(ǫ) (1 − ǫ)E[e2] + ǫE[y2], 0 ≤ ǫ ≤ 1

• 2. Minimize H(ǫ) over all stabilizing controllers

inf

K stabilizing P H(ǫ) H∗(ǫ).

• 3. The optimal tracking error H∗

e satisfies

H∗

e ≥

1 1 − ǫ (H∗(ǫ) − ǫΓ) , ∀0 ≤ ǫ ≤ 1

• 4. Because of the convexity of the objective and constraint functionals

H∗

e = sup 0≤ǫ≤1

• 1

1 − ǫ (H∗(ǫ) − ǫΓ)

• .

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel

Approach

inf

K stabilizing P E[e2], subject to E[y2] ≤ Γ

• 1. Define

H(ǫ) (1 − ǫ)E[e2] + ǫE[y2], 0 ≤ ǫ ≤ 1

• 2. Minimize H(ǫ) over all stabilizing controllers

inf

K stabilizing P H(ǫ) H∗(ǫ).

• 3. The optimal tracking error H∗

e satisfies

H∗

e ≥

1 1 − ǫ (H∗(ǫ) − ǫΓ) , ∀0 ≤ ǫ ≤ 1

• 4. Because of the convexity of the objective and constraint functionals

H∗

e = sup 0≤ǫ≤1

• 1

1 − ǫ (H∗(ǫ) − ǫΓ)

• .

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel

Approach

inf

K stabilizing P E[e2], subject to E[y2] ≤ Γ

• 1. Define

H(ǫ) (1 − ǫ)E[e2] + ǫE[y2], 0 ≤ ǫ ≤ 1

• 2. Minimize H(ǫ) over all stabilizing controllers

inf

K stabilizing P H(ǫ) H∗(ǫ).

• 3. The optimal tracking error H∗

e satisfies

H∗

e ≥

1 1 − ǫ (H∗(ǫ) − ǫΓ) , ∀0 ≤ ǫ ≤ 1

• 4. Because of the convexity of the objective and constraint functionals

H∗

e = sup 0≤ǫ≤1

• 1

1 − ǫ (H∗(ǫ) − ǫΓ)

• .

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel

Approach

inf

K stabilizing P E[e2], subject to E[y2] ≤ Γ

• 1. Define

H(ǫ) (1 − ǫ)E[e2] + ǫE[y2], 0 ≤ ǫ ≤ 1

• 2. Minimize H(ǫ) over all stabilizing controllers

inf

K stabilizing P H(ǫ) H∗(ǫ).

• 3. The optimal tracking error H∗

e satisfies

H∗

e ≥

1 1 − ǫ (H∗(ǫ) − ǫΓ) , ∀0 ≤ ǫ ≤ 1

• 4. Because of the convexity of the objective and constraint functionals

H∗

e = sup 0≤ǫ≤1

• 1

1 − ǫ (H∗(ǫ) − ǫΓ)

• .

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 7 / 17

Tracking Performance Over an Additive White Noise Channel

Preliminaries

Coprime factorization P = NM−1 = ˜ M−1 ˜ N All stabilizing two-parameter controllers: K = {K : K =

• K1

K2

• = (˜

X−R˜ N)−1×

• Q

˜ Y − R ˜ M

• , Q, R ∈ RH∞}

The power can be expressed in terms of the H2 norm as E[e2] = (I − NQ) ψr2

2 + TΘ2 2

E[y2] = NQψr2

2 + TΘ2 2

T I − N(˜ X ˜ Nr − R) ˜ M Θ diag(

• Φ1, . . . ,
• Φm)

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 8 / 17

Tracking Performance Over an Additive White Noise Channel

Allpass Factorization

Factorize ˜ MΘ ˜ MΘ: ˜ MΘ(s) = ˜ M(m)

Θ (s)˜

BΘ(s), where ˜ M(m)

Θ

is the outer factor of ˜ MΘ and ˜ BΘ = ˜ B(Np)

Θ

˜ B(Np−1)

Θ

· · · ˜ B(1)

Θ ,

with ˜ B(i)

Θ (s) = I − 2Re{pi}

s + ¯ pi ζiζH

i .

The pole direction vector ζi: ζi =

• ζ(1)

i

, . . . , ζ(m)

i

T , ζH

i ζi = 1

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 9 / 17

Tracking Performance Over an Additive White Noise Channel

The Optimal Tracking Performance

Theorem 1 Assume that P(s) has simple unstable poles pi ∈ C+, i = 1, . . . , Np. Define η 2

Np

• i=1

pi

m

• j=1

|ζ(j)

i |2Φj.

Then the system is stabilizable if and only if Γ > η and the best tracking performance under the channel input power constraint E[y2] ≤ Γ is H∗

e =

  

• σ2

r − √Γ − η

2 + η if η < Γ < η+σ2

r

η if Γ ≥ η+σ2

r

σ2

r = E[r(t)Tr(t)]: reference input power.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 10 / 17

Tracking Performance Over an Additive White Noise Channel

The channel input power

Total channel input power E[y2] =

• Γ,

if η < Γ < η+σ2

r ,

η + σ2

r ,

if Γ ≥ η+σ2

r .

Power allocation of the parallel channel The power distributed to the kth sub-channel is             

Γ−η σ2

r σ2

rk + 2 Np

• i=1

pi|ζ(k)

i

|2Φk, if η < Γ < η+σ2

r ,

σ2

rk + 2 Np

• i=1

pi|ζ(k)

i

|2Φk, if Γ ≥ η+σ2

r .

The channel is not exploited to the maximum extent allowable.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 11 / 17

Tracking Performance Over an Additive White Noise Channel

Power Allocation vs Water Filling

Water Filling for Transmission (Shannon)

Maximize the capacity Pi = (ν − Ni)+, X (ν − Ni)+ = P

Power Allocation for Tracking:“Fire Quenching”

Minimize the tracking error 8 > > > > < > > > > :

Γ−η σ2

r σ2

rk + 2 Np

X

i=1

pi|ζ(k)

i

|2Φk, if η < Γ < η+σ2

r ,

σ2

rk + 2 Np

X

i=1

pi|ζ(k)

i

|2Φk, if Γ ≥ η+σ2

r . NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 12 / 17

Joint Channel/Controller Design

Joint Channel/Controller Design: Scaling

[K1 K2] P + Λ Λ−1 n(t) r(t) y(t) yλ(t)

Λ = λI: uniform scaling for all channels. The optimal design problem: inf

λ>0,K∈K E[e2], subject to E[yλ2] ≤ Γ

K: the set of all stabilizing controllers.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 13 / 17

Joint Channel/Controller Design

Optimal Tracking Performance

Theorem 2 The best tracking performance under the channel input power constraint E[yλ2] ≤ Γ is given by J∗

e = σ2 r

η Γ, when the scaling factor is chosen as λ∗ =

Γ

√

σ2

r (Γ−η). The corresponding total

channel input power is E[yλ2] = Γ. Furthermore, the power allotment for the kth channel is Γ − η σ2

r

σ2

rk + 2 Np

• i=1

pi|ζ(k)

i

|2Φk. The joint design of the controller and the scaling factor improve the performance.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 14 / 17

Joint Channel/Controller Design

SISO Systems

Corollary 3 The system is stabilizable if and only if

Γ Φ > 2

Np

• i=1

pi.

Under this condition, the best tracking performance is

J∗

e = σr 2

• 2

Np

• i=1

pi

• Φ

Γ .

The tracking performance in this case is inversely proportional to the channel SNR.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 15 / 17

Joint Channel/Controller Design

Comparison

Tracking over an AWN channel

[K1 K2] P + n(t) r(t) y(t)

H∗

e =

8 < : “p σ2

r − √Γ − η

”2 + η if η < Γ < η+σ2

r ,

η if Γ ≥ η+σ2

r .

Perfect tracking can never be achieved.

Tracking over an AWN channel with scaling

[K1 K2] P + Λ Λ−1 n(t) r(t) y(t) yλ(t)

J∗

e = σ2 r

η Γ Perfect tracking can be achieved when the power constraint Γ is allowed to be infinite.

Tracking a step signal

[K1 K2] P u(t) r(t) y(t)

Perfect tracking can always be achieved for minimum phase plants.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 16 / 17

Summary

Summary

The problem:

• 1. Optimal tracking of MIMO control systems over AWN parallel channels

with input power constraint.

• 2. Optimal power allocation for best tracking.

Results

• 1. Analytical expression of the best achievable tracking performance

exhibiting a clear dependence on channel power, SNR, etc, and additionally on the plant unstable poles.

• 2. Optimal power allocation strategy for optimal tracking performance.
• 3. While in the ideal case it is possible to achieve perfect tracking for

minimum phase systems, it can never be done with a power-constrained channel.

• 4. The performance can be improved by using a simple scaling scheme,

which can be interpreted as a simple pre-compensation of the feedback channel.

NSF Workshop in Honor of BH Performance of Networked Feedback System October 4, 2010 17 / 17