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Discretize Continuous State Space Model

HardMath, Signal Processing 📚 Advanced Architectures

Implement a discretization method for State Space Models originally defined in continuous time to accommodate discrete token sequences. This process is crucial for applying continuous-time models to real-world problems that involve discrete data.

The Continuous State Space Model is defined by the equations $x'(t) = Ax(t) + Bu(t)$ and $y(t) = Cx(t)$ , where $x(t)$ is the state vector, $u(t)$ is the input vector, and $y(t)$ is the output vector. To discretize this model, we can use the Zero-Order Hold method, which assumes the input is constant between sampling times.

Here are the steps to discretize the model:

Define the continuous-time model parameters $A$ , $B$ , and the sampling time $\Delta$ .
Compute the discretized state transition matrix $\bar{A}$ and input coefficient $\bar{B}$ .

\bar{A} = e^{A \cdot \Delta}

\bar{B} = \frac{e^{A \cdot \Delta} - 1}{A} \cdot B

This technique is widely used in digital control systems.

Example:

Input:

A=-1.0, B=1.0, delta=0.1

Output:

(0.9048, 0.0952)

Reasoning:

A_bar = e^(-0.1) ≈ 0.9048. B_bar = (e^(-0.1) - 1) / (-1) × 1 ≈ 0.0952. The state decays by ~10% each step while accumulating input.

Constraints:

$A$ : Continuous state transition (scalar), $-10 \leq A \leq 0$
$B$ : Input coefficient (scalar), $0 \leq B \leq 10$
$\Delta$ : Discretization step size, $0.01 \leq \Delta \leq 2.0$

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Discretize Continuous State Space Model

Background Knowledge

State Space Models originate from continuous-time dynamical systems in control theory. To use them for discrete sequences (text, images), we must discretize the continuous equations.

Continuous-Time SSM

The continuous system is defined by:

dh/dt = Ah(t) + Bx(t) # State derivative
y(t) = Ch(t) # Output

Where:

$h(t) \in \mathbb{R}^S$ is the hidden state
$A \in \mathbb{R}^{S \times S}$ is the state transition matrix
$B \in \mathbb{R}^{S \times 1}$ is the input projection
$C \in \mathbb{R}^{1 \times S}$ is the output projection

Zero-Order Hold (ZOH) Discretization

The Zero-Order Hold method assumes the input $x(t)$ is constant between samples. This yields:

Ā = exp(ΔA)
B̄ = (exp(ΔA) - I) A⁻¹ B = (Ā - I) A⁻¹ B

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