Key Sequence

Notation

• Linear Dynamical System
• set operations

New Concepts

• reachability analysis
• Set Propagation Techniques

Important Results / Claims

Questions

Set Propagation Techniques

Now, how exactly do we compute the reachable set.

…for Linear Dynamical Systems

Let’s consider writing \(s’\) as a function of \(s, x\). For a linear system, we have:

\begin{equation} s’ = \qty(T_{s}+T_{a}\Pi_{o}O_{s})s + T_{a}\Pi_{o}x_{o} + T_{a}x_{a} + x_{s} \end{equation}

our goal is to then write this for sets of initial \(s\). Write this in terms of Set Propagation Techniques:

Key Sequence

Notation

New Concepts

Important Results / Claims

• overapproximate
• inclusion functions

Questions

Interesting Factoids

reachability for non-linear systems

Standard reachability analysis for Linear Dynamical System is not great, because polytopes don’t stay polytopes when we apply non-linear operations.

The general vibe, then, is to take a non-linear thing and bound them using a polytope.

interval arithmetic

We can’t propagate polytopes though non linear systems; but we can propagate intervals.

Suppose we have an interval:

\begin{equation} [x] = \qty {x \mid x_1 \leq x \leq x_2} \end{equation}

Key Sequence

Notation

New Concepts

Important Results / Claims

• Taylor Models
• Concrete Reachability

Questions

Interesting Factoids

Partitioning

As a way to prevent wrapping effect, you can partition your initial set into smaller chunks and propagate them separately, and then union them together.

Discrete Reachability

Remember: we can represent discrete systems using directed graphs.

reachable sets for discrete systems

Breadth-first search to get the reachable sets. We can terminate the BFS by checking if one subset of points is contained within another:

Key Sequence

Notation

New Concepts

• explainability
  • policy visualization
  • feature importance
  • integrated gradients
  • shapley values
• Surrogate Models
• counterfactual

Important Results / Claims

Questions

Interesting Factoids

Alignment Problem

autonomous systems will do exactly what we tell them to do… so we need to give them good instructions. This is the Alignment Problem

1. imperfect objective—underspecified objective
2. imperfect model—understanding of the world is underspecified
3. imperfect optimization—the model just didn’t solve the problem correctly

Validation Framework

High level structure:

validation_algorithm(system, spec)

system

• environment: state of the world, \(T(s’|s,a)\)
• sensor, \(O(o|s)\)
• agent, policy \(\pi\qty(a | o)\)

example: inverted pendulum

• state: \(\qty (\theta, \omega)\) of the pendulum
• observation: \(O(o|s) = \mathcal{N}\qty (o|s,\Sigma)\), Gaussian noise
• policy: consider the following proportional controller policy

\begin{equation} \pi \qty(a | o) = \begin{cases} 1, \text{if} a = -15 \tau - 8 \omega \\ 0 \end{cases} \end{equation}