State Estimation Task

STATE-ESTIMATION-TASK contains a single data file (data.csv) which itself contains inputs and outputs from a dynamical system (see below). The task is to estimate the unobserved states of this dynamical system from the inputs and outputs.

Dynamics

State Transition Model

$$x_{t+1} = 0.5 x_t + \frac{3.0 x_t}{1 + x_t^2} + u_t + w_t$$

where:

• $x_t$ is the state at time $t$
• $u_t$ is the control action at time $t$
• $w_t \sim \mathcal{N}(0, \sigma_w^2)$ is the process noise with $\sigma_w = 0.3$

Observation Model

$$y_t = 0.1 x_t^2 + v_t$$

where:

• $y_t$ is the observation at time $t$
• $v_t \sim \mathcal{N}(0, \sigma_v^2)$ is the observation noise with $\sigma_v = 0.2$

Initial Conditions

• Initial state: $x_0 = 0.5$
• Control input: $u_t = \sin(10\pi t / T)$ where $T$ is the total number of time steps
• Time horizon: $T = 100$ steps

Task

Given the inputs (actions) $u$ and outputs (observations) $y$ in data.csv, produce an estimate of the unobserved states. You may use any state estimation method that you deem to be effective.