Linear System (LinearSystem-v0)

The Linear System is the simplest benchmark environment and the canonical test bed for LQR and SKVI. The dynamics are globally linear, so LQR delivers the exact optimal solution and provides an unambiguous performance ceiling.

Dynamics

The discrete-time state transition is

$$x_{t+1} = A \, x_t + B \, u_t$$

where $A \in \mathbb{R}^{3 \times 3}$ is a random stable matrix (largest absolute real eigenvalue $< 1$, drawn once at environment construction) and $B = \mathbf{1}_{3 \times 1}$.

Spaces and limits

Property	Value
State dimension	3
Action dimension	1
State range	$[-25, 25]^3$
Action range	$[-10, 10]$
Max episode steps	200
Integration	Direct (discrete map)

Cost function

$$c(x, u) = x^\top Q x + u^\top R u, \qquad Q = I_3, \quad R = I_1$$

Reference point: $x^* = \mathbf{0}$.

Usage

uv run -m koopmanrl.soft_koopman_value_iteration --env_id LinearSystem-v0
uv run -m koopmanrl.linear_quadratic_regulator --env_id LinearSystem-v0

Notes

• A is randomised at construction. Each new LinearSystem() instance draws a fresh $A$ matrix, so seeds must be fixed at the process level for reproducibility.
• LQR is exact here. SKVI and SAKC results on this environment should be compared against the LQR solution as an upper bound.
• Source: koopmanrl/environments/linear_system.py