SI&DA - Lecture 15 'System Identification'

Professor Notes

notes_system_identification.pdf

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System Identification

Why do we need to find a model of the system? → Control the System → Prediction

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~Ex.:

Given the input i obtain the following output

→ Model of the System

Will be something like: $y(t)=4\cdot u$ → f(u) = $4\cdot u$

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~Ex.:

Same input as before but the output is slightly different (There is a delay in the input/output response)

→ $y(t)=4\cdot u(t-1)$

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~Ex.:

Same input, another output, this time really different from the output.

The solution this time around is: $y(t)=-3\cdot u(t-1)+2\cdot u(t-2)$

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~Ex.:

Strangely the solution is the same as before: $y(t)=-3\cdot u(t-1)+2\cdot u(t-2)$

This happens because of the noise affecting the system

→ The first 2 examples where noise-free, while in this example there is a noise. → We don’t know the noise, it immeasurable.

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Noise

Assuming the noise is a Random Process that it’s affecting the output we can model the system as: Where: → $u(t)$ : input → $y(t)$ : output → $e(t)$ : noise

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Main Elements of System Identification

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Data Set

Characteristic of the Data Set

• Data Set Domain: in this course we will consider input/output data in the time domain (another possibility could be the frequency domain)
• Length of the Data Set
• Sampling Time

The input $u$ can be chosen by the user or not. ~Ex.: in an electrical circuit i can choose the input voltage and current (can be chosen by the user) ~Ex.: the input of the system is given by a controller (cannot be chosen by the user)

The input must be as informative as possible → The second definition of the input is much more meaningful

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Model Class: Parametric System Identification

What we will see is a kind of system identification called Parametric System Identification, this technique has the scope of finding the parameters that define a given model:

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~Ex.: Family of Transfer Functions

Then i can say that the “parameter vector” $\theta$ : We want to find specific values of $K$ and $\alpha$, our parameters, we call this specific values $\theta$ the parameter vector. While the family of all possible parameter vectors in this case $\left[K\in \mathbb{R},\alpha \in \left[0,2\pi \right]\right]$ is called $\Theta$ also called a priori-information.

Or i can say that:

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Model Class: Non-Parametric Models

For example i want to identify a model by its impulse response Or maybe I’m interested in its frequency response

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Model Class: Gray or Black Box Model

If we have some prior information on the system. ~Ex.: if we have some physical information about the system. Then we can use the “Gray Box Model”.

Else if we don’t have prior information we talk about “Black Box Model”.

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~Ex.: Spring Dump Mass Model

We know the formula of the spring and dump, also the first law of Newton so we can use the Gray Box Model.

NOTE: We can instead use the Black Box Model but we will throw away useful prior information, using only input and output response.

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Choice of the ‘best’ model

We want to find ${\hat{\theta }}^{\ast }$ which minimizes a given cost function $J(\theta )$

The complexity depends on various elements:

• $J$ : the function i want to minimize
• $\dim (\theta )$ : how big is theta

Suppose you have a set of candidate models $\mathcal{M}$, then I want to choose a model inside $\mathcal{M}$ such that its “distance” is minimum from the theoretical model given by the input-output model (given for example by a Black Box Model).

NOTE: We say this because often no model inside $\mathcal{M}$ respects the input-output model, and remember we can only chose a model inside $\mathcal{M}$.

→ So given a set of model $\mathcal{M}$ we need to validate them.

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Model Validation

Assess the quality of the model and it capacity to satisfy the requirements. → It must be able to predict (or simulate) future outputs

Given a system $S$ and the model $M$ we want to minimize the error $\epsilon (t)$

Reasons to fail validation:

• Optimization process may not converge ~Ex.: due to numerical process (cannot find a feasible solution)
• Wrong choice of the cost function $J$
• Wrong choice of the model class $\mathcal{M}$ ~Ex.: System of second order, we choose a first order model class
• The dataset is not informative enough

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Identification Process

An iterative process that I need to repeat until I find a good model. → A good practice is also to iterate even if I find a good model at first try.

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Principle of Parsimony

If i have 2 models that have same quality and same performance I prefer the simplest one.

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