Point-to-Point Real-Time Communication Over a Backplane Bus

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Introduction to estimation theory
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Contents

• 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.Introduction
1.1 What is estimator for signal models (1/1)
Parameter estimation
Parameter
estimator
State
State estimation
as small as possible
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1.Introduction
1.2 estimator application (1/3)

• Other methodology
• Fault detection
• parameter estimation
• state observer/estimation
• signal separation
• spectrum analysis
• Output feedback control state feedback control
  estimator

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1.Introduction
1.2 estimator application (2/3)
Output feedback control state feedback control
estimator
plant
Control
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1.Introduction
1.2 estimator application ( 3/3)

• 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

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1.Introduction
1.3 Signal models (1/3)

• Categories of signal models

Time invariant
Discrete-time
State space
Stochastic
Modeled
Linear
Nonlinear
Unmodeled
Time varying
Deterministic
Generic linear model
Continuous-time
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1.Introduction
1.3 Signal models (2/3)

• State space signal model

• In case of stochastic model

are random process
and

• In case of deterministic model

are deterministic signal
and

• Choice of model is important for model-based
  signal processing

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1.Introduction
1.3 Signal models (3/3)

• Modelled vs unmodelled signal

• Unmodeled signal

• Model based signal

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1.Introduction
1.4 Design objectives (1/1)

• Stability of the filter
• Estimation error ( often called performance
  ) unbiasedness convergence
  efficiency
• Robustness estimation error w.r.t signal model
  uncertainties
• Computation load

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1.Introduction
1.5 Options for estimators (1/1)
Performance Criterion
generic linear
minimax
Signal Models
infinite horizon
Minimum variance
stochastic
state space
least square
deterministic
receding horizon
IIR (infinite horizon)
FIR (receding horizon)
Given
Options
Initial state dependent
Initial state independent
Estimator structure
Nonlinear
Linear
Smoothing
Prediction
Filter
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1.Introduction
1.6 Objectives and design procedure (1/2)
Yes
No
Signal models
Optimal estimator
desired properties
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1.Introduction
1.6 Objectives and design procedure (2/2)
Objectives
Options
Stability

• FIR

Small error

• Performance criterion

• Robustness
• w.r.t uncertainties
• w.r.t disturbance

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1.Introduction
1.7 Options for estimator smoothing,
filtering, and predicting

• Categories of estimators

Current time
Smoothing
Filtering
Predicting
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1.Introduction
1.8 FIR structure (1/ 9)

• Case 1 (IIR)

• Case 2 (FIR)

Which one do you think better ?
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1.Introduction
1.8 FIR structure (2/9)

• BIBO stability of FIR estimators

Case 1 (FIR)
Case 1 (IIR)
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1.Introduction
1.8 FIR structure (3/9)

• Robustness to model uncertainty

Divergence of IIR filter (Kalman filter)
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1.Introduction
1.8 FIR structure (4/9)

• Robustness to round off error comparison of
  error covariance

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

• 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.

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1.Introduction
1.8 FIR structure (5/9)

• Deadbeat property

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

In case of Control

1. Stabilization the nominal system
2. Stabilization the disturbed systems

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1.Introduction
1.8 FIR structure (6/9)

• Deadbeat property

• Horizon size

Noise
State estim. trajectory
Exact filter (deadbeat phenomenon)
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1.Introduction
1.8 FIR structure (7/9)

IIR filter
FIR filter
Original
Time
Filtered
Magnitude
Heavy distortion of phase at band gap
Frequency
Phase
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1.Introduction
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

cf. Infinite impulse response(IIR)

• Nonlinear phase
• Not always stable
• Easy to obtain from analog filter
• Suitable for sharp cutoff characteristic and
  high speed

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1.Introduction
1.8 FIR structure (9/9)
F I R
I I R
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1.Introduction
1.9 Initial state dependency (1/2)

• Infinite impulse response (IIR)

dependent of
Linear
Initial state dependent
IIR
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1.Introduction
1.9 Initial state dependency (2/2)

• Filter is to estimate stateThe initial state
  is also a state It is not logical
  to assume the initial state

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1.Introduction
1.10 Performance criterion (1/3)

• Performance criterion

Maximum Likelihood
Minimum variance
Least square
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1.Introduction
1.10 Performance criterion (2/3)

• Performance criterion for deterministic models-
  filter- Minimax filter-
  Least squares

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1.Introduction
1.10 Performance criterion (3/3)
Objectives
Options
Stability

• FIR

• Minimization

Small error

• Robustness
• w.r.t uncertainties
• w.r.t disturbance

• Minimization of maxima

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1.Introduction
1.11 Extension to control receding horizon
control
What is the receding horizon control?
Which one do you think better ?
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1.Introduction
1.11 Extension to control desired property

• Stability of the closed-loop systems
• Small tracking error
• Robustness stability tracking error

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1.Introduction
1.11 Extension to control options for controls
Performance Criterion
I/O model
Signal Models
minimax
stochastic
LQG
infinite horizon
state space
LQ
deterministic
receding horizon
Given
Options
output feedback
state feedback
Finite memory control(including static control)
Dynamic(IIR control)
Control structure
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1.Introduction
1.11 Extension to control objectives and design
procedures
Desired properties Stability Robustness Small
tracking error
Yes
No
Signal models
Optimal control
desired properties
Performance criterion LQG LQ Minimum entropy
Control structure
State feedback control Output feedback
controlDynamic controlFinite memory control
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1.Introduction
1.11 extension to control performance criterion
with receding horizon

• LQ

• LQG

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1.Introduction
1.11 extension to control receding horizon
output feedback control

• method 1

Filter Kalman filter filter Mixed filer
State feedback receding horizon control LQC
Control

Question Is it optimal ?

• method 2

Global optimal output feedback control
FMC (finite memory control)
cf) LQG
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1.Introduction
1.11 extension to control receding horizon
output feedback control

• Computation

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1.Introduction
Contents of standard textbook on optimal control
and estimation

• 1. LQ control
• Finite horizon
• Infinite horizon
• 2. Kalman filter
• Finite horizon
• Infinite horizon
• 3. LQG control
• Finite horizon
• Infinite horizon

• 4. Full information control
• Finite horizon
• Infinite horizon
• 5. filter
• Finite horizon
• Infinite horizon
• 6. Output feedback control
• Finite horizon
• Infinite horizon

Covered in this class
Seoul Natl Univ.