SI&DA - Lecture 16 'LTI Models'

Linear Time Invariant (LTI) Models

Knowing that:

Then:

• And by the probability density function $f_e(e)$ of $e(t)$

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Parametric System Identification

If we assume that the noise is Gaussian also the pdf is known.

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ARX Model

Auto Regressive with eXoguenous inputs → Similar to an AR Model

Expect for this part:

NOTE: The AR model is defined as: $y(t)+a_{1}y(t-1)+\dots +a_{n_a}y(t-n_a)=e(t)$

We define:

So the ARX Model can be re-written as:

$$A(z)\cdot y(t)=B(z)\cdot u(t)+e(t)$$

And i can find that:

And I can represent the model as:

The parametric vector model $\theta$ is given by:

$$\theta =[a_1,\dots ,a_{n_a},b_1,\dots ,b_{n_b}{]}^T$$

Of dimensions: $\dim (\theta )=[(n_a+n_b)\times 1]\triangleq [d\times 1]$

(This strong assumptions is difficult to motivate)

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FIR Models

Finite Impulse Response

• Subset ofTODO [ARX Model]

$$\begin{array}{l}{n}_a=0\\ \Rightarrow A(z)=1\end{array}$$

Then the input/output response will be equal to:

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Monic Polynomial

$C(z)$ is defined as monic if it’s “leading coefficient” (the …) is $1$ and all the other coefficient are multiplied by $z^{-1}$, $z^{-2}$, $\dots$, or $z^{-n}$ (resulting in a polynomial of grade 0)

~Ex.:

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ARMAX Model

TODO [Auto Regressive Moving Average] with eXtrogenous Inputs

We define:

So:

Model representation

NOTE: $B(z)$ is notTODO [monic] Because the system does not depend on the input at time $t$ : $u(t)$ The input at time $t$ is just the noise $e(t)$

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OE Model

Output Error

We assume that

Where $w(t)$ is defined as: → Which is very similar to anTODO [ARMAX Model], except for the noise that is moved to the formula above

We define:

So: → Which is different from anTODO [ARX Model] because the error $e(t)$ is not filtered.

The structure of an OE model is defined as follows:

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BJ Model

Box-Jenkins Model, the name of this model is given after the 2 surnames of the guys that firstly used this model.

We define:

Where $D(z)$ is aTODO [Monic Polynomial]:

The model structure is:

The parametric vector model $\theta$ is given by: → Differently from theTODO [OE Model] is that we have new parameters: $c_1,\dots ,c_n$ and $d_1,\dots ,d_n$ This means having a more complex model, that results in: → More freedom → The identification process is more complex, so more prone to numerical errors

So for theTODO [Principle of parsimony] if I can represent well my system with a more simple model (in this case theTODO [OE Model]), I should do so.

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General Model Class

Let’s group all the following model classes into one:

• TODO [ARX Model]
• TODO [FIR Model]
• TODO [ARMAX Model]
• TODO [OE Model]
• TODO [BJ Model]

So for:

• $A=F=C=D=1$ → We obtain anTODO [FIR Model] (just $B(z)\ne 1$)
• $F=C=D=1$ →TODO [ARX Model] ($A(z)$ and $B(z)$ $\ne 1$)
• $F=D=1$ →TODO [ARMAX Model] ($A(z)$, $B(z)$ and $C(z)$ $\ne 1$)
• $A=C=D=1$ →TODO [OE Model] ($B(z)$ and $F(z)$ $\ne 1$)
• $A=1$ →TODO [BJ Model] ($B(z)$, $C(z)$, $D(z)$ and $F(z)$ $\ne 1$)

NOTE: For theTODO [Principle of parsimony] if I can represent the general system with a more simple model (one from the list above), I should do so.

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Delay in Input

If we know the delay $r$ we the problem is the same asTODO [General Model Class]. Otherwise we have a new parameter to estimate, the delay $r$.