Introduction to estimation theory Seoul Nat’l Univ.

Introduction to estimation theory

C o n tro l In fo rm a tio n S ys tem L a b .Seoul Nat’l Univ.

Contents

C o n tro l In fo rm a tio n S ys tem L a b .

What is estimator for signal models estimator application Signal models Design objectives Options of estimators Objectives and design procedure Options for estimator : smoothing, filtering, and predicting FIR structure Initial state dependency Performance criterion Extension to Control

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1.1 What is estimator for signal models (1/1)

1.Introduction


Parameter estimation

State estimationestimator: Parameter

: State

as small as possible

Other methodology Fault detection

parameter estimation

state observer/estimation

signal separation

spectrum analysis

Output feedback control : state feedback control + estimator


?

?)(

tx

1u2u

3u 4u

1y2y

3y

),,,,,,()(ˆ 3214321 yyyuuuuftx

1.2 estimator application (1/3)

1.Introduction

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Control

Output feedback control = state feedback control + estimator

1.Introduction1.2 estimator application (2/3)

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plant

estimator

Practical areas Speech

- speech enhancement Image

- medical imaging- denoising

aerospace - target tracking- navigation- flight pass reconstruction

chemical process- distillation columns

mechanical system - motor system

biological area- cardiac arrhythmia detection


1.Introduction1.2 estimator application ( 3/3)

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Modele

d

Unmodeled

State s

pace

Generi

c linea

r model

Linear

Nonlinea

r

Stochast

ic

Determ

inistic

Time i

nvarian

t

Time v

arying

Discret

e-tim

e

Continuous-t

ime

Categories of signal models


1.Introduction1.3 Signal models (1/3)

State space signal model

In case of stochastic model :

In case of deterministic model :

and are random process

and are deterministic signal Choice of model is important for model-based signal processing



Modelled vs unmodelled signal

velocity

VoltageInput :

velocityOutput :



Unmodeled signal

Model based signal

Stability of the filter Estimation error ( often called performance )

unbiasedness : convergence :

efficiency : Robustness

estimation error w.r.t signal model uncertainties Computation load


1.Introduction1.4 Design objectives (1/1)

Estimator structure

Performance

CriterionSignal

Models

IIR

(infinite horizon)

FIR

(receding horizon)

Initial state

dependent

stochastic

deterministicleast square

minimax


1.Introduction

Given OptionsInitial state

independent

Minimum

variance

NonlinearLinear

1.5 Options for estimators (1/1)

FilterSmoothing Prediction

generic linear

state space

receding horizon

infinite

horizon

Signal models Optimal estimator Does it satisfy

Estimator structure

. . .

Performance criterion

. . .

desired properties

Desired properties

Stability

Robustness

Small error Yes

No


1.Introduction

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1.6 Objectives and design procedure (1/2)

Stability FIR

1.Introduction

Small error

Robustness w.r.t uncertainties

w.r.t disturbance


Objectives : Options

1.6 Objectives and design procedure (2/2)

1.Introduction1.7 Options for estimator : smoothing, filtering, and predicting


Categories of estimators

Current time

Smoothing Filtering Predicting


Which one do

you think better ?

Case 1 (IIR)

Case 2 (FIR)

1.Introduction

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1.8 FIR structure (1/ 9)


Case 1 (FIR)

Case 1 (IIR)

1.Introduction

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1.8 FIR structure (2/9)

BIBO stability of FIR estimators

0 100 200 300 400 500 600 700 800 900 1000-80

-70

-60

-50

-40

-30

-20

-10

0

10Real state Estimate state Diverged estimate state

Divergence of IIR filter (Kalman filter)


1.Introduction1.8 FIR structure (3/9)

Robustness to model uncertainty


Robustness to round off error : comparison of error covariance

Observation : Though rounding at the 4th digit are not serious, rounding of 3rd and 2 nd digit makes

difference between the FIR filter and IIR filter.

FIR filter Kalman Filter

4th digit 0.018184 0.02188

3rd digit 3.019853 12.3217

2nd digit 4.240283 15.47567

Round-off digitFilter Structure

5.0,5.0

01

1

1

1.0

1.0

9950.00998.0

0998.09950.01

RQ

vxy

wuxx

kkk

kkkk

Simulation environments We assume that the filter gain is previously known by off-line calculation Rounding off error is applied when updated Model

1.Introduction

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1.Introduction

Seoul Nat’l Univ. C o n tro l In fo rm a tio n S ys tem L a b .

1. Stabilization the nominal system2. Stabilization the disturbed systems

In case of Control

Nominal systems

1. Be sure to be deadbeat using FIR structure for nominal systems.

2. Small error for noise or disturbance corrupted systems.

In case of Filter


Require to be deadbeat using nominal systems Nominal systems = zero disturbance / noise system

Deadbeat property

1.Introduction



)(tx

)(ˆ tx

Exact filter (deadbeat phenomenon)

Noise

State & estim. trajectory

Horizon size

Deadbeat property


Original

Filtered

IIR filter

1.Introduction

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0 0.2 0.3 1-1

0

1Chebyshev-1 Lowpass Filter

frequency (pi units)

PH

AS

E

0 0.2 0.3 1-1

0

1FIR Lowpass Filter


PH

AS

E

0 0.2 0.3 1

50

0

FIR Lowpass Filter


Ma

gn

itu

de

|H

(w)|

in

dB

0 0.2 0.3 1

50

0.25 0

Chebyshev-1 Lowpass Filter


Ma

gn

itu

de

|H

| in

dB

Magnitude

Phase

Time

Frequency

FIR filter

Heavy distortion of

phase at band gap


cf. Infinite impulse response(IIR) : Nonlinear phase Not always stable Easy to obtain from analog filter Suitable for sharp cutoff characteristic and high speed

1.Introduction

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1.8 FIR structure (8/9) Advantage & disadvantage

• Advantage of FIR Use of DFT Robustness to round off error Linear phase Guaranteed stability Good for adaptive filter

• Disadvantage of FIR Computation load H/W complexity


F I RF I R I I R

1.Introduction

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Infinite impulse response (IIR) : dependent of

IIR Linear Initial state dependent

1.Introduction

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1.9 Initial state dependency (1/2)

FIR Linear Initial state free

Finite impulse response (FIR) : Independent of

Filter is to estimate stateThe initial state is also a state

It is not logical to assume the initial state


1.Introduction

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1.9 Initial state dependency (2/2)

Example : Try to guess who he is.

1. First case(our approach)

2. Second case(ex. Kalman filter) ,

Given, then guess

Given, then guess

Original picture


Minimum variance

Least square

Maximum Likelihood


1.Introduction1.10 Performance criterion (1/3)

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Performance criterion for deterministic models- filter

- Minimax filter

- Least squares

H



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Performance criterion for Stochastic models- Minimum variance

- Minimax variance

- Minimum Entropy



Stability FIR

Small error

Robustness w.r.t uncertainties

w.r.t disturbance

Minimization

Objectives : Options

Minimization of maxima

1.Introduction1.11 Extension to control : receding horizon control


Which one do

you think better ?

What is the receding horizon control?

1.Introduction


Stability of the closed-loop systems Small tracking error Robustness

stabilitytracking error

1.11 Extension to control : desired property

Control structure

Performance

CriterionSignal

Models

infinite horizon

receding horizon

output feedback

stochastic

deterministic LQ

minimax

Given Optionsstate feedback

LQG

Finite memory control(including static control)

Dynamic(IIR control)

I/O model

state space


1.Introduction1.11 Extension to control : options for controls

1.Introduction


Signal models Optimal control Does it satisfy

Control structure

Performance criterionLQG

LQ

Minimum entropy

……

desired properties

Desired properties

Stability

Robustness

Small tracking error Yes

No

State feedback control

Output feedback controlDynamic controlFinite memory control

……

1.11 Extension to control : objectives and design procedures

1.Introduction


1.11 extension to control : performance criterion with receding horizon

LQ

LQG

1.Introduction1.11 extension to control : receding horizon output feedback control


State feedback receding horizon controlLQC Control……

FilterKalman filter filterMixed filer……

+

Question : Is it optimal ?

FMC (finite memory control)

method 1

method 2 Global optimal output feedback control

cf) LQG

1.Introduction


Computation

1.11 extension to control : receding horizon output feedback control

Contents of standard textbook on optimal control and estimation

1.Introduction


1. LQ control Finite horizon Infinite horizon

2. Kalman filter

Finite horizon Infinite horizon

3. LQG control


4. Full information control Finite horizon Infinite horizon

5. filter


6. Output feedback control


Receding horizon

Receding horizon

Receding horizon

Receding horizon

Receding horizon

Receding horizon

Covered in this class

Introduction to estimation theory Seoul Nat’l Univ.

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Transcript of Introduction to estimation theory Seoul Nat’l Univ.