Learn the Unknown Part of a Physical Model

Learning Can Begin with a Physical Model

BOTIE

An engineer may know the form of a machine's motion without knowing its coefficients accurately. A motor's response can differ from its nominal specification, and a controller built around the wrong response will make poor predictions. Measuring inputs beside the resulting outputs gives us evidence about the missing parameters.

Åström and Bohlin's 1966 work on identifying linear dynamic systems studied learning such models from operating records. System identification is this task of estimating a system's behaviour from observations. It connects a chosen physical model with statistical estimation instead of requiring every coefficient to be supplied in advance.

We will estimate just one gain through the origin using least squares. The model says which relationship to fit; the recorded observations determine its coefficient. An offset or a delay would change the model we need, so a good fit to this restricted example cannot establish that every physical effect has been captured.

Sources for this history

We Know the Relationship, but Not the Gain

Botie has a simple model of a conveyor: over a fixed short interval, displacement is approximately proportional to command. The proportionality depends on the actual drive and load. We need to learn it from measurements.

This is system identification. Rather than fitting an unrestricted function immediately, we state a model structure and estimate its unknown parameter. That structure gives us assumptions we can test.

Finite aligned measurements with nonzero command energy; no actuator is driven. Python handles missing excitation explicitly.

Step 1 of 1

Measure the response to different commands

Each displacement uses the same observation interval.

Changing commands gives information about gain; zero commands alone do not.
Command	Displacement
1	2
2	4
3	6

1
Displacement
2
2
Displacement
4
3
Displacement
6

Changing commands gives information about gain; zero commands alone do not.

Choose the model and what is observed

Least squares balances residuals under the declared proportional model.

Fit the conveyor’s unknown gain

Botie knows a proportional model but needs the parameter for this measured system.

BOTIE

Botie knows a proportional model but needs the parameter for this measured system.

Choose an example

Measured commands

1

2

3

Measured displacements

1

2

3

Estimated command-to-displacement gain

Follow the operations

1. Measured commands
1. 01
2. 12
3. 23
2. Measured displacements
1. 02
2. 14
3. 26
3. Dot product
```
28
```
4. Dot product
```
14
```
5. Divide
```
2
```
6. Result
```
2
```

Finite aligned measurements with nonzero command energy; no actuator is driven. Python handles missing excitation explicitly.

1. Choose the model and what is observed

Let u be command and y displacement over a fixed interval. The model predicts y as gain times u. Subtract that prediction from the measured displacement to get a residual, the part this model did not explain. The gain has units of displacement per command.

Assume commands are measured accurately enough and residuals do not systematically depend on them. Noise in commands or feedback-correlated disturbances can bias a simple least-squares fit.

Keep the interval fixed or include it explicitly. Mixing durations changes the relationship being fitted.

2. Minimise squared residuals for the single gain

Differentiate the sum of squared prediction errors with respect to gain and set the derivative to zero. The result is a ratio of two sums.

gain_hat = sum(u × y) / sum(u × u)
u = [1,2,3], y = [2,4,6]
gain_hat = (2 + 8 + 18) / (1 + 4 + 9) = 2

A zero denominator means the experiment supplied no nonzero command. The parameter is unidentified by these observations; return an error rather than inventing a gain.

Check whether the command supplied information

Command energy is the denominator of the through-origin estimate.

Python

import numpy as np
commands = np.array([0.,0.])
print(float(commands @ commands))

Output

0.0

These commands cannot identify the gain. The correct response is an explicit failure, not division by zero or an arbitrary parameter.

Check yourself

Did the experiment excite the system?

Question 1 of 2

What gain is fitted?

Use the information below.

Commands [1,2], displacement [3,6]. Model y=g*u.

3. A fitted parameter is conditional on the experiment

Try commands over the intended operating range. If the actuator saturates at larger commands, a single gain may fit small commands and fail elsewhere. Inspect residuals against command and time.

Choose experiments within physical limits. More excitation can improve information while also increasing wear or risk; this numerical lesson is not an instruction to probe real hardware without a safety plan.

Training fit is not validation. Keep different operating conditions for an independent check.

4. Use the model to explain a failed prediction

If errors grow with load, consider whether load belongs in the model. If errors lag behind command changes, consider dynamics rather than only a gain.

A learned residual model can augment a physical model, but it should be evaluated on the conditions where the combined model will be used.

The next lesson keeps several conditions visible so an aggregate score cannot hide a transfer failure.

5. Weak excitation can amplify measurement error

A denominator need not be exactly zero to be a problem. If commands are tiny relative to measurement noise, a small response perturbation can change the fitted gain substantially. Repeat the estimate across plausible measurement perturbations and inspect uncertainty.

For multiple parameters, columns that vary together create a related identifiability problem. More data repeating the same uninformative experiment may help far less than an experiment that separates those effects.

The nonzero check prevents an undefined calculation. It does not certify that the parameter is precisely estimated.