Fundamental Limitations in Networked Control Systems and - - PowerPoint PPT Presentation

Fundamental Limitations in Networked Control Systems and Quantization

Hid ki I hii Hideaki Ishii

Tokyo Institute of Technology Tokyo Institute of Technology

ishii@dis.titech.ac.jp HYCON2 PhD S h l C t l f HYCON2 PhD School on Control of Networked and Large-Scale Systems June 22, 2011

Networked control: The other perspective ?

Control

Communi- cation

Information technology & communication Traditionally, for connecting humans & computers Very mature area

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Networked control: The other perspective

Control

Communi- cation

Networked control is new in communication also!! For real-time control: Machine to machine Hi h li bilit d b t f i l t k High reliability and robustness for wireless networks E g : Factory automation Medical devices E.g.: Factory automation, Medical devices Standardization for different layers

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Standardization for different layers

Example: Automotive LAN

In-vehicle electronic devices connected via networks Protocols for real-time control: CAN, FlexRay,…‧

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Channel capacity considerations

NCS: Constraints on channel capacity Even if the total capacity is large, each component may use only a (small) portion New challenges: Modeling of the capacity constraints in NCS Modeling of the capacity constraints in NCS Allocating bandwidth to each transmission g What is the necessary capacity for control?

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Motivating example: Inverted pendulum g p p

Given a sampling period 10 ms p g p Angle data is quantized: E.g. 8 bits 800 bps For stabilization, how much precision in bits is needed?

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In NCS, two fields meet

Communication / Information theory Transmission of data generated by random processes Not concerned with the content of the info and delay Not concerned with the content of the info and delay Control theory Control theory Information is specific, used for feedback control Traditionally, assumed infinite bandwidth Stochastic vs deterministic Stochastic vs deterministic

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Sahai & Mitter (2006), Nair, Fagnani, Zampieri, & Evans (2007)

Interplay between control & communication

Two directions:

• 1. Control over networks with capacity constraints
• 2. Systems analysis based on information

theoretic tools We will observe some fundamental limitations We will observe some fundamental limitations arising in feedback control with networks Analogous to Shannon’‚s source coding theory

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Outline

• 0. Introduction
• 1. Control under capacity constraints

a The minimum data rate for stabilization

• a. The minimum data rate for stabilization
• b. The coarsest quantization for stabilization
• 2. Information theoretic approach to Bode’‚s

integral formula integral formula

• 3. Conclusion

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• 1. Control under capacity constraints

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Quantization and coding

In control systems, signals take real values

g

y , g To be transmitted over finite capacity channels, they must be transformed to discrete values

Quantizer Channel Quantizer & Encoder Decoder

Assume no error/delay in the channel Focus on quantization

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Quantized control: Problem setup

Plant Plant Controller Controller Channel Quantizer

Plant Discrete-time, LTI ( ) Unstable, but stabilizable (or controllable)

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Uniform quantizer q

Quantization error:

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General quantizer q

if Partition of Output values Index set : finite/countable Index set : finite/countable Are there quantizer structures suitable for control? Are there quantizer structures suitable for control?

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Quantized control

Plant Plant Controller Controller Channel Quantizer

Control objective: Stabilization Fundamental question: How much information is needed from the quantized signal to achieve this objective?

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Control with quantized signals q g

Traditionally, quantized error is modeled as additive y, q white noise This model becomes inaccurate when quantization l ti i resolution is coarse

noise

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Bertram (1958), Curry (1970)

Delchamps’‚ observation 1 p

State info via uniform quantizer: q Result: Result: Assume is stable. Apply the control There is a region around the origin into which each trajectory goes. (= Practical stability) Inside there, behavior is chaotic.

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Delchamps (1990)

Delchamps’‚ observation 2 p

Result: If the plant is not so unstable as then there is control h h h j such that each trajectory goes to 0. Quantized info may be sufficient for precise control!

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Outline

• 0. Introduction
• 1. Control under capacity constraints

a The minimum data rate for stabilization

• a. The minimum data rate for stabilization
• b. The coarsest quantization for stabilization
• 2. Information theoretic approach to Bode’‚s

integral formula integral formula

• 3. Conclusion

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• 1a. The minimum data rate for stabilization

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Control under capacity constraint p y

Plant Plant Controller Controller Channel Quantizer

How small can the data rate be for stabilization? Total # of discrete values = D t t [bit / l ] Data rate = [bits/sample] First studied by Wong & Brockett (1999)

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First studied by Wong & Brockett (1999)

Minimum data rate: Scalar case

Plant

intervals into Partitions N ] 1 , 1 [ :

• 1
• 1

Corresponding control inputs control inputs

be? should large how , within keep To N k x ] 1 , 1 [ ) (

• 22

Minimum data rate: Scalar case

Theorem Theorem

• le]

[bits/samp rate Data

• le]

[bits/samp rate Data

• Minimum data rate depends on the unstable pole

More unstable plants require more data rate

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Wong & Brockett (1999)

Minimum data rate: Scalar case

Let ) (

Proof

satisfy must width its Then, hand

• ther

the On y , Th hand,

• ther

the On Thus,

1

• 1

1 1 let , [

• n

quantizer uniform a With ) ( ] 1 , 1

• Then

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Minimum data rate: General case

Necessary part

Plant

cells into Partitions

• f

subset Bounded N : : cells into Partitions N :

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Wong & Brockett (1999)

Minimum data rate: General case

Necessary part

Plant

cells into Partitions

• f

subset Bounded N : : cells into Partitions N :

Theorem

• Rate

Data

[bits/sample] : Unstable eigenvalues of

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Wong & Brockett (1999)

Dynamic quantizer y q

Time-varying quantizers with memory Example: Digital camera When the state is outside the range, zoom out

Zoom out

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Brockett & Liberzon (2000)

Dynamic quantizer y q

When the state is inside the range, hold and then apply control T l t th t t i l i To locate the state more precisely, zoom in

Zoom in

Global asymptotic stabilization may be achieved

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Additional 2 bits: Zoom in/out, Hold

Minimum data rate: General setup

Plant Plant Controller Controller

Encoder

Channel Encoder

Encoder

# of code words

Average data rate Controller

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Controller

The minimum data rate

Theorem Theorem as

le] [bits/samp rate data Ave

• General results: Control is impossible if the bound

d t h ld! does not hold!

Deterministic case: Tatikonda & Mitter (2004) Stochastic case: Nair & Evans (2004)

P f b t ti Proof by construction:

Quantizer transmission scheme controller

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Quantizer, transmission scheme, controller

Structure of the controller

Plant Plant Controller One-step Controller ahead Dynamic Q Estimator Estimator

Channel

: Coarse estimate of state from quantized signal

Channel

) ( ˆ k x

: Coarse estimate of state from quantized signal : Estimate of one-step ahead

) (k x

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At the encoder

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At the encoder

The overall quantizer

x R L E c x Q ˆ ) , , , , (

• x

ˆ x

4 states Determine region of

i

R

2

2R

2

x

Determine region of quantization Shared by encoder

2

l

c

Shared by encoder and decoder

• : Bits assigned to

2

e

2

1 2

1

2R R

• : Bits assigned to

the ith mode

1

e

1

l

1 1

2

u

R

• log
• i

R

1 1

x

u i i

R

• 2

log

• 33

Comments and further studies

Average data rate

[bits/sample]

Stochastic control case General class of noise and disturbance

Stabilization in the mean-square sense Stabilization in the mean-square sense The same minimum rate: Does not depend on noise statistics

Nair & Evans (2004), Tatikonda et al. (2004), Matveev & Savkin (2004), Yuksel & Basar (2006), Ishii, Ohyama, & Tsumura (2008)

With multiple sensors & controllers

( ) y ( ) Tatikonda (2003), Yuksel & Basar (2007)

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Outline

• 0. Introduction
• 1. Control under capacity constraints

a The minimum data rate for stabilization

• a. The minimum data rate for stabilization
• b. The coarsest quantization for stabilization
• 2. Information theoretic approach to Bode’‚s

integral formula integral formula

• 3. Conclusion

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• 1b. The coarsest quantization for stabilization

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Quantized control: Memoryless case y

Dynamic quantizers May require large memory and heavy computation b th d f th h l

• n both ends of the channel

Memoryless quantizers Example: Uniform quantizer Example: Uniform quantizer The level of quantization error is the same for any input. I it ll t b fi h ? Is it really necessary to be so fine everywhere?

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The coarsest quantizer for stabilization q

To achieve stability via Lyapunov approach, the “”coarsest”„ quantization can be characterized Fine around the origin, but coarse outside Coarseness is determined by plant unstable poles Characterization of another fundamental limitation

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Elia & Mitter (2001), Fu & Xie (2005), Ishii & Francis (2002)

Stabilization via coarse quantizer q

Pl Plant Controller Quantizer Channel

Plant Control input

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Stabilization via coarse quantizer q

Plant Controller Quantizer Channel

Quantizer: Memoryless = Piecewise constant function Channel Noiseless No error/delay Channel Noiseless, No error/delay Objective: Quadratic stabilization

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j

Quadratic stabilization

Given a matrix , let Find and such that is a Lyapunov function Find and such that is a Lyapunov function for the quantized closed-loop system: Th f th t d t h ti The energy of the system decreases at each time

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Discrete control inputs

Given Set of states for which V decreases with Set of states for which V decreases with

where

Recall the control is Recall the control is

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Discrete control inputs

To choose so that there is no gap, we should take for some

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The “”coarsest”„ quantizer

Among the quantizers that can stabilize the plant wrt

q

g q p the Lyapunov function , the “”coarsest”„ is logarithmic.

) (v Q

Parameter:

) (v Q

Larger = Coarser

v

2

• 44

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Elia & Mitter (2001)

Optimization of coarseness p

Find the maximum value of coarseness over all quadratic Lyapunov functions: Th Theorem

Plant unstable poles

The corresponding matrix is the solution to

Plant unstable poles

The corresponding matrix is the solution to to the Riccati equation and the feedback gain is

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and the feedback gain is

Limitation in quantization q

Coarse

Stabilization region Coarseness in Coarseness in quantization

Fine

Level of plant instability Level of plant instability

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Extensions: In the presence of uncertainties p

Plant Plant Controller Controller Quantizer Channel

Two cases:

• 1. In the channel: With packet losses
• 2. In the plant: Adaptive control

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Extension 1: With erasure channel

Plant Controller Quantizer Channel

Uncertainty in the channel Random packet losses

Quantizer Channel

Objective: Stability in a stochastic sense We will find limitations in quantization and packet loss

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Tsumura, Ishii, & Hoshina (2009)

Quantized control over erasure channel

Plant Controller Quantizer Channel

Channel model

Loss process: IID Loss probability

If lost If lost If received

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Stochastic quadratic stability

Given a quadratic function Find , , and s.t. stochastic quadratic stability is guaranteed:

Lyapunov function in a stochastic sense y p

Sufficient for mean-square stability

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When only packet loss is present y p p

Critical bound on loss rate for mean-square stability. q y

Stabilization region Loss Stabilization region probability Level of plant instability

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Sinopoli et al. (2004), Elia (2005), Imer, Yuksel, & Basar (2006), Ishii (2009)

Limitation in quantization and packet loss

Theorem

q p

Theorem There exists a control law to achieve stochastic quadratic stabilization L b bilit Loss probability L ith i ti Logarithmic quantizer

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Limitation in quantization and packet loss

6

Coarseness in

q p

4 5

quantization

2 3 1 2 1 2 3 0.2 0.4 0.6 0 8

Packet loss prob. L l f l t

3 4 5 6 0.8 1.0

Level of plant instability

For unstable plants, we need reliable and fine info

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Numerical example

Plant Unstable poles Loss prob. , where Proposed method Conventional method

Larger in magnitude State x2 Control u

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Time Time

Extension 2: Quantized adaptive control

Plant Controller Uncertainty in the plant: Unknown parameters in Quantizer Channel Adaptive control law: Ti i f db k i d l i h i i Time-varying state feedback gain and logarithmic quantizer Guarantees closed-loop Lyapunov stability in : p y p y and boundedness in

Hayakawa, Ishii, & Tsumura (2009), Siami, Hayakawa, Ishii, & Tsumura (2010)

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Adaptive quantization

Coarseness in : Depends on the feedback gain In general, when the gain in large, quantization must be fine. Quantization in the coarseness is introduced.

) (v Qk ) (v Qk

v

• k
• 2

k

• v

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Adaptive quantization

Generalization of Elia & Mitter (2001) ( )

Same result if system matrices are completely known.

When the plant is uncertain

Th ti t t th t ( hi h i The quantizer may not converge to the coarsest (which is determined by the unstable poles). This is because the time-varying gain may not equal the ti l ( ft it )

• ptimal one (even after it converges).

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Numerical Example

Plant

Parameter for coarseness

Uncertain parameters

where

State Ad ti i Adaptive gain C t l i t Coarseness parameter Control input parameter Ti Ti

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Time Time

15 values Final value: 1.67

Outline

• 0. Introduction
• 1. Control under capacity constraints

a The minimum data rate for stabilization

• a. The minimum data rate for stabilization
• b. The coarsest quantization for stabilization
• 2. Information theoretic approach to Bode’‚s

integral formula integral formula

• 3. Conclusion

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• 2. Information theoretic approach to

Bode’‚s integral formula

Relation between two theories

S t Input Output System

Information theory Control theory We may expect complementary relation by considering a problem in one field (Control) from the viewpoint of the other Information the other Information Info theory: Less stringent assumptions on systems

–— Bode’‚s integral formula: From LTI to nonlinear

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Zhang & Iglesias (2003)

Bode’‚s integral formula

L: Discrete time, LTI, SISO, Strictly proper y p p

• Assume that the closed-loop is stable.

Sensitivity function:

Unstable poles of L

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Bode (1945), Freudenberg & Looze (1985), Sung & Hara (1989)

Fundamental limitation: Water-bed effect

Unstable poles of L Unstable poles of L

Not possible to achieve arbitrary sensitivity reduction e g

• ver the whole freq range

reduction, e.g., over the whole freq. range Based on the transfer function and complex analysis p y (Jensen formula)

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Bode’‚s via an information theoretic approach

Sensitivity case

Ratio of power spectral densities

Unstable poles of L

Results by analyzing the entropies of signals

spectral densities

y y g p g Stochastic version of Bode’‚s formula Complementary sensitivity case has been studied also

Martins & Dahleh (2008), Okano, Hara, & Ishii (2009)

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Definitions from information theory

Entropy: Uncertainty in a random variable

: probability density f ti f function of

Mutual information: Info that has about

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Problem setting

L:

• Input is asymptotically stationary Gaussian
• d

i d d t

• and are independent
• Not completely known nor unknown

Not completely known nor unknown The closed-loop is stable

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Problem setting

L: L:

• Sensitivity-like function

–— Ratio of power spectral densities –— In general, it holds that

However, in transfer function,

Derivation in 3 steps

does not hold

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Step 1 PSDs and entropy rates

Entropy rate

y

Power spectral density and entropy rate Power spectral density and entropy rate Equality holds if x is Gaussian System and entropy rates of its I/O

Difference in entropy rates of I/O log| Ratio of PSDs |

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py

Step 2 Conservation law of entropy

How does affect ?

M t l i f b t th i it t t Mutual info between the init state and output

Then, divide this by k and take limsup

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Step 3: Relation to unstable poles

If the closed loop is stable, then

• Implication: As the poles become more unstable,

signals in the loop contain more info about

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Bode’‚s via an information theoretic approach

Th

pp

Theorem Assume

• and

are independent

• is asymptotically stationary Gaussian
• is asymptotically stationary Gaussian
• th

l d l i t bl the closed loop is stable Then,

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In a general NCS setting

• Controller K: Nonlinear system, required to be causal

Networked control setup Networked control setup Encoder/decoder may be included in K y Exogenous input c: Channel noise Limitations in sensitivity wrt channel capacity

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Martins & Dahleh (2008), Ishii, Okano, & Hara (2011)

Conclusion

NCS with channel capacity considerations y Quantized control The minimum data rate C i i Coarse quantization Fundamental limitations related to both Fundamental limitations related to both control and communication!! More recently: Quantization in multi-agent systems

Kashap, Basar, & Srikant (2007), Carli, Fagnani, Speranzon, & Zampieri (2008), Cai & Ishii (2011)

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