Identification

6.435

SET 6 – Parametrized model structures – One-step predictor –

Munther A. Dahleh

Lecture 6 6.435, System Identification 1 Prof. Munther A. Dahleh Models of LTI

• A complete model

u = input y = output e = noise (with PDF).

Lecture 6 6.435, System Identification 2 Prof. Munther A. Dahleh • A parametrized model

u = input y = output e = noise (with PDF of e). e white noise

Lecture 6 6.435, System Identification 3 Prof. Munther A. Dahleh One Step Linear Predictor

•

• Want to find that minimizes

•

• Minimum Prediction Error Paradigm

Data:

Estimate of at time N

Lecture 6 6.435, System Identification 4 Prof. Munther A. Dahleh One Step Linear Predictor (Derivation) The predicted output has the form

Both & have a delay. Past inputs and outputs are mapped to give the new predicted output.

Notice that:

Lecture 6 6.435, System Identification 5 Prof. Munther A. Dahleh Now:

has at least has at least one delay one delay

The lower bound is achieved if z = 0

Equivalently

Lecture 6 6.435, System Identification 6 Prof. Munther A. Dahleh Result

Lecture 6 6.435, System Identification 7 Prof. Munther A. Dahleh Examples of Models

–ARX (Autoregressive with exogenous input) • Description

•Matched with the model

Lecture 6 6.435, System Identification 8 Prof. Munther A. Dahleh Examples … ARX

•One step predictor

•

(a function of past )

• Prediction error

Lecture 6 6.435, System Identification 9 Prof. Munther A. Dahleh Examples …. ARMAX

–ARMAX (Autoregressive with exogenous input) • Description

• Standard model

• More general, includes ARX model structure.

Lecture 6 6.435, System Identification 10 Prof. Munther A. Dahleh Examples …. ARMAX

•One step predictor

or

• Pseudo-linear Regression past predictions

where or simply Not linear in

Lecture 6 6.435, System Identification 11 Prof. Munther A. Dahleh Examples …. OE

• OE (Output Error) • Description

•One step predictor

•Standard

Lecture 6 6.435, System Identification 12 Prof. Munther A. Dahleh Examples …. OE

• Vector

Define

Lecture 6 6.435, System Identification 13 Prof. Munther A. Dahleh Examples …. Box-Jenkins

An even more general model

Lecture 6 6.435, System Identification 14 Prof. Munther A. Dahleh Examples …. State-Space

–State-Space Models • Description

disturbance •Noise: two components Output noise Usual assumptions

w, v are white.

Lecture 6 6.435, System Identification 15 Prof. Munther A. Dahleh • One Step Predictor =

is a positive semi-definite solution of the steady-state Riccati equation:

(error )

Lecture 6 6.435, System Identification 16 Prof. Munther A. Dahleh Innovation Representation

In general

Lecture 6 6.435, System Identification 17 Prof. Munther A. Dahleh General Notation

Define

It follows

Describes the model structure

Predictor:

Lecture 6 6.435, System Identification 18 Prof. Munther A. Dahleh For a given

• Notice: can be stable even though is not!

•

Lecture 6 6.435, System Identification 19 Prof. Munther A. Dahleh Predictor Models

Def: A predictor model is a linear time-invariant stable filter that defines a predictor

Def: A complete probabilistic model of a linear time-invariant system is a pair of a predictor model and the PDF associated with the prediction error (noise).

In most situations, is not complete known. We may work with & .

Lecture 6 6.435, System Identification 20 Prof. Munther A. Dahleh Stability Requirements

Example ARX

If has zeros outside the disc, then the map from is unstable.

is always stable.

Lecture 6 6.435, System Identification 21 Prof. Munther A. Dahleh Model Sets

Def: A model set is a collection of models

Index set

Examples all linear models all models where is FIR of fixed order a finite set of models nonlinear fading memory models

Comment: These are “big” sets that are not necessarily parametrized in a nice way.

Lecture 6 6.435, System Identification 22 Prof. Munther A. Dahleh Model Structures

Model structures are parametrizations of model sets. We require this parametrization to be smooth.

Let a model be index by a We require to be differentiable with respect to

Lecture 6 6.435, System Identification 23 Prof. Munther A. Dahleh Notice that

Def: A model structure is a differentiable map from a connected open subset to a model set such that the gradient, is defined and stable.

is the map is one particular model

Lecture 6 6.435, System Identification 24 Prof. Munther A. Dahleh Example: ARX model structure

stable

Lecture 6 6.435, System Identification 25 Prof. Munther A. Dahleh General Structure

Lecture 6 6.435, System Identification 26 Prof. Munther A. Dahleh You need

to be differentiable

is stable

(not sufficient).

Lecture 6 6.435, System Identification 27 Prof. Munther A. Dahleh Model Structures

Proposition: The parametrization

(= set of of A , B , C , D , F )

restricted to the set

has no zeros outside the unit disc

is a model structure

Lecture 6 6.435, System Identification 28 Prof. Munther A. Dahleh Proof: Follows from the previous general derivative. Notice that H may be unstable!

Proposition: The Kalman Filter parametrization is a model structure if

(The stability property is equivalent to being detectable).

Lecture 6 6.435, System Identification 29 Prof. Munther A. Dahleh Independent Parametrization

A model structure m is independently parametrized if

Useful for giving a domain interpretation of the estimate.

• We can define a model set as the of a model structure:

• We can define unions of different model structures:

Useful for model structure determination!

Lecture 6 6.435, System Identification 30 Prof. Munther A. Dahleh Identifiability

Central question I: Does the identification procedure yield a unique parameter ? Data Identification procedure Informative Identifiability Question

Def: Model structure is identifiable (globally!) at if It is locally identifiable at if there exists an such that implies

Def: Model structure is strictly identifiable (local or global) if it is identifiable (local or global) for all

Lecture 6 6.435, System Identification 31 Prof. Munther A. Dahleh Central question II: Is the identified parameter equal to the “true parameters” ?

Parametrized structure: true system

Case I:

Case II:

Define:

Let for some . If is identifiable at , then

Lecture 6 6.435, System Identification 32 Prof. Munther A. Dahleh Identifiability of Model Structures

General:

Identifiable at

ARX:

If &

&

ARX is strictly identifiable

Lecture 6 6.435, System Identification 33 Prof. Munther A. Dahleh OE:

Suppose . Then

& iff (B, F ) are coprime.

(To do this cleanly, need to consider where & are the delay powers in both B & F )

Lecture 6 6.435, System Identification 34 Prof. Munther A. Dahleh Identifiability

Theorem:

is identifiable at iff

1) There are no common factors of 2) ” 3) ”

Lecture 6 6.435, System Identification 35 Prof. Munther A. Dahleh