Uncertainty Models

This page describes the uncertainty models used to generate (random) disturbances or sensor noise for closed-loop simulation. To enable seamless interchangeability of different uncertainty models (e.g., Gaussian, uniform distributions), a common interface including a dataclass for generated uncertainty vectors are provided. The implemented concrete models are (briefly) presented at the end of this file.

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Dataclass Objects

Generated uncertainty vectors are stored as instances of the corresponding UncertaintyInterface dataclass. Each vehicle model must implement its concrete disturbance dataclass.

The uncertainty dataclass objects can be converted to numpy arrays using their convert_to_array() method.

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Overview and Interface Concept

All uncertainty models implement the same abstract interface, UncertaintyModelInterface.

Let the uncertainty be represented by a random vector \(z \in \mathbb{R}^d\) with fixed dimension \(d\). Each uncertainty model provides:

• a nominal value $$ \bar{z} = \mathbb{E}[z] \quad \text{(or a user-defined value)} $$

• a sampling operator $$ w \sim \mathcal{Z} $$ where \(\mathcal{Z}\) denotes the underlying distribution.

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Interface Definition

Required Parameters

Parameter	Variable type	Description
dim	int	Dimension of the uncertainty vector
nominal_value	np.ndarray/List[float]/UncertaintyInterface	User-defined nominal value or default value defined in the concrete child class

Required Methods

Method	Description
nominal_value()	Returns the nominal value
sample_uncertainty()	Draws a random sample from the underlying uncertainty distribution

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Implemented Uncertainty Models

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No Uncertainty

Dummy uncertainty model if, for instance, no disturbances or sensor noise is desired.

Default nominal value: \(\bar{z} = 0\)

Additional parameters:

No additional parameters required.

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Gaussian Distribution

Models a normally distributed random variable, i.e., \(z \sim \mathcal{N}(\mu,\sigma^2)\) with mean \(\mu \in \mathbb{R}^d\) and standard deviation \(\sigma \in \mathbb{R}^d\).

Default value: \(\bar{z} = \mu\)

Additional parameters:

Parameter	Variable type and default value	Description
mean	np.ndarray/List[float]/UncertaintyInterface	Mean
std_deviation	np.ndarray/List[float]/UncertaintyInterface	Standard deviation

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Uniform Distribution

Models a uniformly distributed random variable \(z\) whose value lies between certain bounds \(a \in \mathbb{R}^d\), \(b \in \mathbb{R}^d\) with \(a \leq b\).

Default nominal value: \(\bar{z} = \frac{1}{2}(a + b)\)

Additional parameters:

Parameter	Variable type and default value	Description
lower_bound	np.ndarray/List[float]/UncertaintyInterface	Lower bound
upper_bound	np.ndarray/List[float]/UncertaintyInterface	Upper bound