SYSTEMS Identification

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SYSTEMSIdentification

• Ali Karimpour
• Assistant Professor
• Ferdowsi University of Mashhad

Reference System Identification Theory For The
User Lennart Ljung
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Lecture 11
Recursive estimation methods

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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Introduction

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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Introduction
In many cases it is necessary, or useful, to have
a model of the system available on-line.
The need for such an on-line model is required in
order to

• Which input should be applied at the next
  sampling instant?

• How should the parameters of a matched filter
  be tuned?

• What are the best predictions of the next few
  output?

• Has a failure occurred and, if so, of what type?

Adaptive control, adaptive filtering, adaptive
signal processing, adaptive prediction.
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Introduction
The on-line computation of the model must
completed during one sampling interval.
Identification techniques that comply with this
requirement will be called

• Recursive identification methods.

• Recursive identification methods. Used in this
  Reference.

• On-line identification.

• Real-time identification.

• Adaptive parameter estimation.

• Sequential parameter estimation.

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Introduction
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Introduction
Algorithm format
This form cannot be used in a recursive
algorithm, since it cannot be completed in one
sampling instant.
Instead following recursive algorithm must comply
Since the information in the latest pair of
measurement y(t) , u(t) normally is small
compared to the pervious information so there is
a more suitable form
Small numbers reflecting the relative information
value in the latest measurement.
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The Recursive Least-Squares Algorithm

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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The Recursive Least-Squares Algorithm
Weighted LS Criterion
The estimate for the weighted least squares is
Where
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The Recursive Least-Squares Algorithm
Recursive algorithm
Suppose the weighting sequence has the following
property
Now
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The Recursive Least-Squares Algorithm
Recursive algorithm
Suppose the weighting sequence has the following
property
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The Recursive Least-Squares Algorithm
Recursive algorithm
Version with Efficient Matrix Inversion
To avoid inverting at each step, let introduce
Remember matrix inversion lemma
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The Recursive Least-Squares Algorithm
Version with Efficient Matrix Inversion
Moreover we have
We can summarize this version of algorithm as
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The Recursive Least-Squares Algorithm
Normalized Gain Version
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The Recursive Least-Squares Algorithm
Initial Condition
A possibility could be to initialize only at a
time instant t0
By LS method
Clearly if P0 is large or t is large, then above
estimate is the same as
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The Recursive Least-Squares Algorithm
Asymptotic Properties of the Estimate
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The Recursive Least-Squares Algorithm
Multivariable case
Remember SISO
Now for MIMO
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The Recursive Least-Squares Algorithm
Kalman Filter Interpretation
The Kalman Filter for estimating the state of
system
The linear regression model
can be cast to above form as
Now, let
Exercise Derive the Kalman filter for above
mention system, and show that it is exactly same
as the Recursive Least-Squares Algorithm for
multivariable case.
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The Recursive Least-Squares Algorithm
Kalman Filter Interpretation
Kalman filter interpretation gives important
information, as well as some practical hints
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The Recursive Least-Squares Algorithm
Coping with Time-varying Systems
An important reason for using adaptive methods
and recursive identification in practice is

• The properties of the system may be time
  varying.

• We want the identification algorithm to track
  the variation.

This is handled by weighted criterion, by
assigning less weight to older measurements
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The Recursive Least-Squares Algorithm
Coping with Time-varying Systems
These choices have the natural effect that in the
recursive algorithms the step size will not
decrease to zero.
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The Recursive Least-Squares Algorithm
Coping with Time-varying Systems
Another and more formal alternative to deal with
time-varying parameters is that the true
parameters varies like a random walk so
Exercise Derive the Kalman filter for above
mention system, and show that it is exactly same
as the Recursive Least-Squares Algorithm for
multivariable case.
Note The additive term R1(t) in P(t) prevents
the gain L(t) from tending to zero.
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The Recursive IV Method

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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The Recursive IV Method
The IV estimate for instrumental variable method
is
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The Recursive IV Method
The IV estimate for instrumental variable method
is
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Recursive Prediction-Error Methods

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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Recursive Prediction-Error Methods
Analogous to the weighted LS case, let us
consider a weighted quadratic prediction-error
criterion
Where
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Recursive Prediction-Error Methods
Analogous to the weighted LS case, let us
consider a weighted quadratic prediction-error
criterion
Remember the general search algorithm developed
for PEM as
For each iteration i, we collect one more data
point, so
now define
As an approximation let
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Recursive Prediction-Error Methods
Analogous to the weighted LS case, let us
consider a weighted quadratic prediction-error
criterion
As an approximation let
With above approximation and taking µ(t)1, we
thus arrive at the algorithm
This terms must be recursive too.
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Recursive Prediction-Error Methods
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Recursive Prediction-Error Methods
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Recursive Prediction-Error Methods
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Recursive Prediction-Error Methods
Family of recursive prediction error methods

• According to the model structure

Wide family of methods

• According to the choice of R

We shall call RPEM
For example, the linear regression
This is recursive least square method
If we consider R(t)I
Where the gain could be normalized so
This scheme has been widely used, under the name
least mean squares (LMS)
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Recursive Prediction-Error Methods
Example 11.1 Recursive Maximum Likelihood
Consider ARMAX model
where
and
Remember chapter 10
This scheme is known as recursive maximum
likelihood (RML)
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Recursive Prediction-Error Methods
Projection into DM
In off-line minimization this must be kept in
mind as a constraint.
The same is true for the recursive minimization.
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Recursive Prediction-Error Methods
Asymptotic Properties
The recursive prediction-error method is designed
to make updates of ? in a direction that on the
average is modified negative gradient of
i.e.
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Recursive Prediction-Error Methods
Asymptotic Properties
Moreover (see appendix 11a), for Gauss-Newton
RPEM, with
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Recursive Pseudolinear Regressions

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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Recursive Pseudolinear Regressions
Consider the pseudo linear representation of the
prediction
And recall that this model structure contains,
among other models, the general linear SISO model
A bootstrap method for estimating ? was given by
(Chapter 10, 10.64)
By Newton - Raphson method
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Recursive Pseudolinear Regressions
By Newton - Raphson method
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Recursive Pseudolinear Regressions
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Recursive Pseudolinear Regressions
Family of RPLRs
The RPLR scheme represents a family of well-known
algorithms when applied to different special
cases of
The RPLR scheme represents a family of well-known
algorithms when applied to different special
cases of
The ARMAX case is perhaps the best known of this.
If we choose
This scheme is known as extended least squares
(ELS).
Other special cases are displayed in following
table
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Recursive Pseudolinear Regressions
Other special cases are displayed in following
table
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The Choice of Updating Step

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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The Choice of Updating Step
Recursive Prediction-Error Methods is based
prediction error approach
Recursive Pseudolinear Regressions is based on
correlation approach
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The Choice of Updating Step
Recursive Prediction-Error Methods (RPEM)
Recursive Pseudolinear Regressions (RPLR)
The difference between prediction error approach
and correlation approach is
We just speak about RPEM, RPLR is the same just
one must change
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The Choice of Updating Step
Update direction
There are two basic choices of update directions
Better convergence rate
Easier computation
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The Choice of Updating Step
Update Step Adaptation gain
An important aspect of recursive algorithm is,
their ability to cope with time varing systems.
There are two different ways of achieving this
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The Choice of Updating Step
Update Step Adaptation gain
In either case, the choice of update step is a
trade-off between

• Tracking ability

• Noise sensitivity

A high gain means that the algorithm is alert in
tracking parameter changes but at the same time
sensitive to disturbance in data.
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The Choice of Updating Step
Choice of forgetting factor
The choice of forgetting profile ß(t,k) is
conceptually simple.
For a system that changes gradually and in a
stationary manner the most common choice is
The constant ? is always chosen slightly less
than 1 so
This means that measurements that are older than
T0 samples are included in the criterion with a
weight e-10.36 of the most recent measurement.
So T0 is the memory time constant.
So we could select ? such that 1/(1-?) reflects
the ratio between the time constant of variations
in the dynamics and those of the dynamics itself.
Typical choices of ? are in the range between
0.98 and 0.995.
For a system that undergoes sudden changes,
rather than steady and slow ones, it is suitable
to decrease ?(t) to a small value and then
increase it to a value close to 1 again.
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The Choice of Updating Step
Choice of Gain ?(t)
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The Choice of Updating Step
Including a model of parameter changes
Kalman Filter Interpretation
Remember
Now let
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The Recursive Least-Squares Algorithm
Kalman Filter Interpretation
In the case of a linear regression model, this
algorithm does give the optimal trade-off between
tracking ability and noise sensitivity, in terms
of parameter error covariance matrix.
The case where the parameters are subject to
variations that themselves are of a nonstationary
nature, i.e. R1(t) varies with t needs a
parallel algorithm. (see Anderson 1985)
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The Choice of Updating Step
Constant systems
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The Choice of Updating Step
Asymptotic behavior in the time-varying case
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Implementation

• Topics to be covered include
• Introduction.
• The Recursive Least-Squares Algorithm.
• The Recursive IV Method.
• Recursive Prediction-Error Methods.
• Recursive Pseudolinear Regressions.
• The Choice of Updating Step.
• Implementation.

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Implementation
Implementation
The basic, general Gauss-Newton algorithm was
given in RPEM and RPLR.
Inverse manipulation is not suited for direct
implementation, (dd matrix R must be inverted)
We shall discuss some aspects on how to best
implement recursive algorithm. By using matrix
inversion lemma
Here ? (a dp matrix) represents either f or ?
depending on the approcach.
But this is pp
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Implementation
Implementation
Unfortunately, the P-recursion which in fact is a
Riccati equation is not numerically sound the
equation is sensitive to round-off errors that
can accumulate and make P(t) indefinite
Using factorization
It is useful to represent the data matrices in
factorized form so as to work with better
conditioned matrices.
Here we shall give some details of a related
algorithm, which is directed based on Householder
transformation (problem 10T.1). (by Morf and
Kailath)
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Implementation
Using factorization

• Step 1 Let

Form (pd)(pd) matrix

• Step 2 Apply an orthogonal (pd)(pd)
  transformation T (TTTI) to L(t-1) so that
  TL(t-1) becomes an upper triangular matrix. (Use
  QR-factorization)

Let to partition TL(t-1) as
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Implementation
Using factorization
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Implementation
Using factorization

• Step 3 Now L(t) and P(t) are

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Implementation
Using factorization in summary
Now L(t) and P(t) are
There are several advantages with this particular
way of performing.

• The only essential computation to perform is
  the triangularization step.

• This step gives new Q and the gain L after
  simple additional calculations.

• Note that ?(t) is triangular pp matrix, so it
  is easy to invert it.

• Condition number of the matrix L(t-1) is much
  better than that of P.