Quantify the Unstable The Main Point Justifications I Li Qiu - - PowerPoint PPT Presentation

▶
quantify the unstable
SMART_READER_LITE
LIVE PREVIEW

Quantify the Unstable The Main Point Justifications I Li Qiu - - PowerPoint PPT Presentation

Sep 15, 2023 395 likes •865 views

Quantify the Unstable The Main Point Justifications I Li Qiu Justifications II Justifications III Conclusions The Hong Kong University of Science and Technology with special thanks to: Collaborators: Jie Chen, Guoxiang Gu,

slide-1
SLIDE 1

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 1

Quantify the Unstable

Li Qiu

The Hong Kong University of Science and Technology

with special thanks to: Collaborators: Jie Chen, Guoxiang Gu, Weizhou Su, Lihua Xie, Ling Shi; Students: Hui Sun, Wei Chen, Shuang Wan, Nan Xiao, Claire Rong Aug 2011

slide-2
SLIDE 2

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 2

Outline

The Main Point Justifications – I Justifications – II Justifications – III Conclusions

slide-3
SLIDE 3

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 3

The Main Point

slide-4
SLIDE 4

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 4

How Bad is an Unstable System?

◮ Which one of the two systems

x(k + 1) = 2x(k) and x(k + 1) = 3x(k), is more unstable?

slide-5
SLIDE 5

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 4

How Bad is an Unstable System?

◮ Which one of the two systems

x(k + 1) = 2x(k) and x(k + 1) = 3x(k), is more unstable?

◮ Which one of the two systems

x(k+1) =

  • 2

1010 4

  • x(k) and x(k+1) =

  3 2 1 3 2 0.5  x(k), is more unstable?

slide-6
SLIDE 6

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 5

Mahler Measure

◮ Consider a polynomial

a(z) = a0zn + a1zn−1 + · · · + an−1z + an = a0

n

  • i=1

(z − ri).

◮ Kurt Mahler in 1960 defined the so-called Mahler measure

M(a) = |a0|

n

  • i=1

max{1, |ri|} = |a0|

  • |ri|>1

|ri|.

slide-7
SLIDE 7

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 5

Mahler Measure

◮ Consider a polynomial

a(z) = a0zn + a1zn−1 + · · · + an−1z + an = a0

n

  • i=1

(z − ri).

◮ Kurt Mahler in 1960 defined the so-called Mahler measure

M(a) = |a0|

n

  • i=1

max{1, |ri|} = |a0|

  • |ri|>1

|ri|.

◮ He also observed that by using Jensen’s formula

M(a) = exp 1 2π 2π log |a(ejω)|dω

  • .

(Geometric mean)

◮ The Mahler measure of a square matrix A

M(A) = M[det(zI − A)].

slide-8
SLIDE 8

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 6

Connection to Szeg¨

  • ’s Problem

◮ For a measurable f : T → C and p ∈ [0, ∞], define

f p = 1 2π 2π |f (ejω)|pdω 1/p f ∞ = lim

pր∞ f p = ess sup ω∈[0,2π]

|f (ejω)| f 0 = lim

pց0 f p = exp

1 2π 2π log |f (ejω)|dω

  • .

◮ M(a) = a0.

slide-9
SLIDE 9

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 6

Connection to Szeg¨

  • ’s Problem

◮ For a measurable f : T → C and p ∈ [0, ∞], define

f p = 1 2π 2π |f (ejω)|pdω 1/p f ∞ = lim

pր∞ f p = ess sup ω∈[0,2π]

|f (ejω)| f 0 = lim

pց0 f p = exp

1 2π 2π log |f (ejω)|dω

  • .

◮ M(a) = a0. ◮ Theorem: (Durand, 1981) Let a be a given polynomial. Then

inf

q aqp = M(a)

where the infimum is taken over all monic polynomials q.

◮ Is this useful?

slide-10
SLIDE 10

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 7

Topological Entropy

◮ Robert Bowen in 1971 defined a quantity h(A) to measure the

complexity and information content of system x(k + 1) = Ax(k), x(k) ∈ Rn.

slide-11
SLIDE 11

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 7

Topological Entropy

◮ Robert Bowen in 1971 defined a quantity h(A) to measure the

complexity and information content of system x(k + 1) = Ax(k), x(k) ∈ Rn.

◮ He called h(A) the topological entropy of A and proved that

h(A) = log M(A).

slide-12
SLIDE 12

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 8

Instability Measure

◮ Main Point: Both M(A) and h(A) can serve as measures of

instability.

slide-13
SLIDE 13

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 8

Instability Measure

◮ Main Point: Both M(A) and h(A) can serve as measures of

instability.

◮ Back to the question in the beginning: For systems

x(k+1) =

  • 2

1010 4

  • x(k) and x(k+1) =

  3 2 1 3 2 0.5  x(k), the latter is more unstable than the former.

slide-14
SLIDE 14

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 9

Justifications – I Difficulty in Stabilization

slide-15
SLIDE 15

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 10

Sensitivity and Complementary Sensitivity

F ✲ ❥ ✲ [A|B] ❄ ✻ v u d x

◮ A

B is a system x(k + 1) = Ax(k) + Bu(k).

◮ Assume v(k) ∈ Rm and

A B is stabilizable.

◮ The sensitivity function S(z) = I + F(zI − A − BF)−1B: the

transfer function from d to u.

◮ The complementary sensitivity function

T(z) = F(zI − A − BF)−1B: the transfer function from d to v.

slide-16
SLIDE 16

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 10

Sensitivity and Complementary Sensitivity

F ✲ ❥ ✲ [A|B] ❄ ✻ v u d x

◮ A

B is a system x(k + 1) = Ax(k) + Bu(k).

◮ Assume v(k) ∈ Rm and

A B is stabilizable.

◮ The sensitivity function S(z) = I + F(zI − A − BF)−1B: the

transfer function from d to u.

◮ The complementary sensitivity function

T(z) = F(zI − A − BF)−1B: the transfer function from d to v.

◮ The smallest achievable norms of S(z) and T(z) capture various

difficulties in stabilizing system A B .

slide-17
SLIDE 17

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 11

The Single-Input Case

◮ Theorem: Let m = 1. For each p ∈ [0, ∞],

inf

F:A+BF is stable S(z)p = M(A).

Furthermore, in the regular case, the optimal F is the same for all p and the optimal S(z) is allpass.

slide-18
SLIDE 18

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 11

The Single-Input Case

◮ Theorem: Let m = 1. For each p ∈ [0, ∞],

inf

F:A+BF is stable S(z)p = M(A).

Furthermore, in the regular case, the optimal F is the same for all p and the optimal S(z) is allpass.

◮ Theorem: Let m = 1.

inf

F:A+BF is stable T(z)2 =

  • M(A)2 − 1

inf

F:A+BF is stable T(z)∞ = M(A).

slide-19
SLIDE 19

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 12 ◮ Special cases of these single-input results appeared in

  • early performance limitation literature (e.g. Sung and Hara,

1988), and

  • recent networked control systems literature (e.g. Elia and Mitter,

2001; Elia, 2005; Fu and Xie, 2005; Braslavsky, Middleton and Freudenberg, 2007).

slide-20
SLIDE 20

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 12 ◮ Special cases of these single-input results appeared in

  • early performance limitation literature (e.g. Sung and Hara,

1988), and

  • recent networked control systems literature (e.g. Elia and Mitter,

2001; Elia, 2005; Fu and Xie, 2005; Braslavsky, Middleton and Freudenberg, 2007).

◮ Naive extension to the multiple-input case does not work.

slide-21
SLIDE 21

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 13

Justifications – II Multivariable Networked Stabilization

slide-22
SLIDE 22

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 14

Multivariable Networked Stabilization

F ✲ ✛ ✚ ✘ ✙ Channels ✲ [A|B] ✻ v u x

◮ What is networked stabilization?

  • involving non-ideal channels.
  • involving channel/controller co-design.

◮ The smallest total “capacity” of the input channels needed so

that the networked stabilization is possible also gives a difficulty in stabilization.

slide-23
SLIDE 23

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 15

Channel Models

✲ ✎ ✍ ☞ ✌ channel ✲ ui vi

◮ What is a communication channel? What is its capacity?

  • Existing information-theoretical model is not very useful.
  • Models capturing individual channel features are available.
  • A timed information theoretical model is needed.
slide-24
SLIDE 24

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 16

Model I: SER Model

✲ ✲ ∆i ❄ ❥ ✲ vi ui ei

◮ ∆i is a nonlinear, time-varying, dynamic uncertain system. ◮ ∆i∞ ≤ δi. ◮ δ−1

i

can be considered as the transmission accuracy or signal-to-error ratio (SER).

◮ Channel capacity Ci = log δ−1

i

= − log δi.

slide-25
SLIDE 25

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 16

Model I: SER Model

✲ ✲ ∆i ❄ ❥ ✲ vi ui ei

◮ ∆i is a nonlinear, time-varying, dynamic uncertain system. ◮ ∆i∞ ≤ δi. ◮ δ−1

i

can be considered as the transmission accuracy or signal-to-error ratio (SER).

◮ Channel capacity Ci = log δ−1

i

= − log δi.

◮ Strongly motivated by logarithmic quantization. (Elia and

Mitter, 2001; Fu and Xie, 2005) ✲ ✻ ✑✑✑✑✑✑ ✑ ✓ ✓ ✓ ✓ ✓ ✓ ✑ ✑ ✑ ✑ ✑ ✑ ✑ ✓ ✓ ✓ ✓ ✓ ✓ vi ui

slide-26
SLIDE 26

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 17

Model II: R-SER Model

✲ ❥ ✲ ✛ ∆i ✻ vi ui ei

◮ ∆i is a nonlinear, time-varying, dynamic uncertain system. ◮ ∆i∞ ≤ δi. ◮ δ−1

i

is received-signal-to-error ratio (R-SER).

◮ Channel capacity Ci = log δ−1

i

= − log δi.

◮ Strongly motivated by an alternative scheme of logarithmic

quantization.

slide-27
SLIDE 27

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 18

Model III: SNR Model

✲ ❄ ❥ ✲ vi ui di

◮ di is zero mean, Gaussian white and vi is zero mean, stationary. ◮ SNRi = σ2

vi

σ2

di

.

◮ Channel capacity Ci = 1

2 log(1 + SNRi).

◮ Strongly motivated by the existing information theory. ◮ Studied in (Braslavsky, Middleton and Freudenberg, 2007)

slide-28
SLIDE 28

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 19

Model IV: Fading Channel

✲ κi(k) ✲ vi ui

◮ κi(k) is an iid process with mean µi and variance σ2

i .

◮ Channel capacity Ci = 1

2 log

  • 1 + µ2

i

σ2

i

  • .

◮ Studied in (Elia, 2005). ◮ Strongly motivated by packet drops, in which case κi(k) is a

Bernoulli process: κi(k) =

  • 1

with probability ρi with probability 1 − ρi In this case, µi = ρi, σ2

i = ρi(1 − ρi), and Ci = − 1 2 log(1 − ρi).

slide-29
SLIDE 29

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 20

Model V: Mixed SER and Packet Drop Model

✲ ✲ ∆i ❄ ❥ ✲ vi κi(k) ✲ ui

◮ ∆i∞ ≤ δi. ◮ κi(k) is a Bernoulli process:

κi(k) =

  • 1

with probability ρi with probability 1 − ρi

◮ Channel capacity Ci = − 1

2 log[ρiδ2 i + (1 − ρi)].

◮ Studied in (Tsumura, Ishii, and Hoshina, 2009)

slide-30
SLIDE 30

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 21

Model VI: Mixed R-SER and Packet Drop Model

✲ ❥ ✲ ✛ ∆i ✻ vi κi(k) ✲ ui

◮ ∆i∞ ≤ δi. ◮ κi(k) is a Bernoulli process:

κi(k) =

  • 1

with probability ρi with probability 1 − ρi

◮ Channel capacity Ci = − 1

2 log[ρiδ2 i + (1 − ρi)].

slide-31
SLIDE 31

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 22

MIMO Channel

✲ ✛ ✚ ✘ ✙ Channels ✲ u v = ✲ ✎ ✍ ☞ ✌ Channel m ✲ um vm . . . ✲ ✎ ✍ ☞ ✌ Channel 1 ✲ u1 v1

◮ A MIMO channel is simply m parallel SISO channels with

possibly different capacities.

◮ The total capacity of all channels

C = C1 + C2 + · · · + Cm.

slide-32
SLIDE 32

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 23

Networked Stabilization

F ✲ ✛ ✚ ✘ ✙ Channels ✲ [A|B] ✻ v u x

◮ One version of networked stabilization involves the design of F

and the allocation of the total capacity to the m channels.

◮ Channel/controller co-design.

slide-33
SLIDE 33

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 23

Networked Stabilization

F ✲ ✛ ✚ ✘ ✙ Channels ✲ [A|B] ✻ v u x

◮ One version of networked stabilization involves the design of F

and the allocation of the total capacity to the m channels.

◮ Channel/controller co-design. ◮ The allocation is given by a probability vector p with

p = p1 · · · pm ′ , 0 ≤ pi ≤ 1, p1 + · · · + pm = 1, such that Ci = piC.

slide-34
SLIDE 34

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 23

Networked Stabilization

F ✲ ✛ ✚ ✘ ✙ Channels ✲ [A|B] ✻ v u x

◮ One version of networked stabilization involves the design of F

and the allocation of the total capacity to the m channels.

◮ Channel/controller co-design. ◮ The allocation is given by a probability vector p with

p = p1 · · · pm ′ , 0 ≤ pi ≤ 1, p1 + · · · + pm = 1, such that Ci = piC.

◮ Networked stabilization problem: Given stabilizable

  • A

B

  • and the total capacity C, find allocation vector p and feedback

gain F so that the closed-loop system is “stable”.

slide-35
SLIDE 35

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 24

Universal Minimum Capacity

◮ Assume that all m input channels are modeled in one of the six

ways above. No mixed modeling.

◮ Universal Minimum Capacity Theorem: The networked

stabilization problem is solvable if and only if C > h(A).

slide-36
SLIDE 36

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 24

Universal Minimum Capacity

◮ Assume that all m input channels are modeled in one of the six

ways above. No mixed modeling.

◮ Universal Minimum Capacity Theorem: The networked

stabilization problem is solvable if and only if C > h(A).

◮ Need to be proved for each of the six channel models.

  • Model I: Gu and Qiu, 2008
  • Model II: Gu and Qiu, 2008
  • Model III: Chen, Qiu, Gu, 2011
  • Model IV: Xiao, Xie, and Qiu, 2009
  • Model V and VI: Wan, Qiu, and Gu, 2011

◮ The proofs are constructive so that an optimal resource

allocation and the corresponding optimal controller F are given.

slide-37
SLIDE 37

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 24

Universal Minimum Capacity

◮ Assume that all m input channels are modeled in one of the six

ways above. No mixed modeling.

◮ Universal Minimum Capacity Theorem: The networked

stabilization problem is solvable if and only if C > h(A).

◮ Need to be proved for each of the six channel models.

  • Model I: Gu and Qiu, 2008
  • Model II: Gu and Qiu, 2008
  • Model III: Chen, Qiu, Gu, 2011
  • Model IV: Xiao, Xie, and Qiu, 2009
  • Model V and VI: Wan, Qiu, and Gu, 2011

◮ The proofs are constructive so that an optimal resource

allocation and the corresponding optimal controller F are given.

◮ May also hold for other channel models.

  • Data rate model.
  • Random delay model.
  • Other mixed models.
  • Timed information theoretical model (to be developed)
slide-38
SLIDE 38

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 25

Justifications – III Output Feedback and Separation Principle

slide-39
SLIDE 39

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 26

Stabilization with Limited Capacity (Rephrase)

F ✲ ✛ ✚ ✘ ✙ Channels ✲ [A|B] ✻ v u x

◮ Now let us consider the packet drop model. ◮ Definition: [A|B] is stabilizable with capacity C if there is an

allocation p and a feedback gain F such that the networked system is stable.

◮ Theorem:

A B is stabilizable with capacity C iff

  • A

B

  • is stabilizable and C > h(A).
slide-40
SLIDE 40

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 27

State Estimation with Limited Capacity

✲ z−1I ✲ C ✲ ✛ ✚ ✘ ✙ Channels ✲ O ✲ ❄ A ✛ x(k + 1) x(k) ˜ x(k) y(k)

slide-41
SLIDE 41

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 27

State Estimation with Limited Capacity

✲ z−1I ✲ C ✲ ❥ ✛ C ✛ z−1I ✛ A ✛ ✲ A ✲ ✻ ✛ ✚ ✘ ✙ Channels ✲ L ❄ ❥ x(k + 1) x(k) ˜ x(k) ˜ x(k + 1) y(k) ˜ y(k) −

◮ Definition:

A C

  • is detectable with capacity C if there is an

allocation p and L such that the e(k) = x(k) − ˜ x(k) ”converges” to zero.

◮ Theorem:

A C

  • is detectable with capacity C iff

A C

  • is

detectable and C > h(A).

slide-42
SLIDE 42

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 28

Output Feedback with limited I/O capacity

✲ A B C D

  • ❄

✛ ✚ ✘ ✙ O Channels I Channels ✛ Controller ✻ ✛ ✚ ✘ ✙ ❄ ❄

◮ Theorem:

A B C D

  • is stabilizable by output feedback with

input capacity Cin and output capacity Cout iff

  • 1. [A|B] is stabilizable with capacity Cin, and

2. A C

  • is detectable with capacity Cout.

◮ Separation principle and observer-based control.

slide-43
SLIDE 43

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 29

Conclusions

◮ M(A) and h(A) give instability measures of matrix A.

slide-44
SLIDE 44

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 29

Conclusions

◮ M(A) and h(A) give instability measures of matrix A. ◮ Bad unstable systems are hard to stabilize and good ones are

easy to stabilize.

slide-45
SLIDE 45

The Main Point Justifications – I Justifications – II Justifications – III Conclusions 29

Conclusions

◮ M(A) and h(A) give instability measures of matrix A. ◮ Bad unstable systems are hard to stabilize and good ones are

easy to stabilize.

◮ In networked stabilization, the channel resource allocation is part

  • f the design. This makes networked control differ from robust

control.