dr-data-driven-predictive-control

name: dr-data-driven-predictive-control description: "Distributionally Robust Data-Driven Predictive Control (DR-DDPC) methodology for stochastic LTI systems with unknown dynamics and disturbance distributions. Combines subspace predictive control (SPC) with distributionally robust optimization using Wasserstein ambiguity sets. Use when: designing robust controllers under uncertainty, implementing data-driven MPC, handling stochastic disturbances with unknown distributions, or applying distributionally robust optimization to control systems."

Distributionally Robust Data-Driven Predictive Control (DR-DDPC)

Overview

A methodology for controlling stochastic linear time-invariant (LTI) systems when both the system dynamics and disturbance distributions are unknown. The approach constructs predictive controllers from input-output trajectory data while providing robustness guarantees against distributional uncertainty via Wasserstein ambiguity sets.

arXiv: 2605.07589
Published: 2026-05-08
Categories: eess.SY, math.OC

Core Methodology

Step 1: Data Collection

Collect a single input-output trajectory {u_t, y_t} from the system:

• System: x_{t+1} = A x_t + B u_t + w_t, y_t = C x_t + D u_t + v_t
• Unknown: A, B, C, D and distributions of w_t, v_t
• Only data required: past input-output pairs

Step 2: Subspace Predictive Control (SPC) Predictor

Fit the SPC predictor via least squares regression:

Y_f = L Z_p + G U_f + E

where:

• Y_f = future outputs horizon
• Z_p = past inputs/outputs (Hankel matrix)
• U_f = future inputs horizon
• L, G = learned predictor matrices
• E = prediction residuals

Step 3: Empirical Residual Distribution

Construct empirical distribution from prediction residuals:

P_N = (1/N) Σ δ_{e_i}

This captures the actual disturbance statistics without parametric assumptions.

Step 4: Wasserstein Ambiguity Set

Define ambiguity set around empirical distribution:

B_ε(P_N) = {Q : W_c(Q, P_N) ≤ ε}

where W_c is the Wasserstein distance and ε controls robustness level.

Step 5: Distributionally Robust Optimization

Solve the DR-MPC problem:

min_{u} max_{Q ∈ B_ε(P_N)} E_Q[cost(u, ξ)]
s.t. chance constraints satisfied ∀ Q ∈ B_ε(P_N)

This ensures performance guarantees even when the true distribution deviates from the empirical one.

Implementation Pattern

import numpy as np
from scipy.optimize import minimize
from sklearn.linear_model import LinearRegression

def build_spc_predictor(u_data, y_data, past_horizon, future_horizon):
    """Build subspace predictive control predictor from data."""
    # Construct Hankel matrices
    N = len(u_data) - past_horizon - future_horizon + 1
    
    Z_p = np.column_stack([
        np.column_stack([u_data[i:i+past_horizon] for i in range(N)]),
        np.column_stack([y_data[i:i+past_horizon] for i in range(N)])
    ])
    
    U_f = np.column_stack([u_data[i+past_horizon:i+past_horizon+future_horizon] 
                           for i in range(N)])
    Y_f = np.column_stack([y_data[i+past_horizon:i+past_horizon+future_horizon] 
                           for i in range(N)])
    
    # Least squares fit
    X = np.vstack([Z_p, U_f])
    coeffs = np.linalg.lstsq(X.T, Y_f.T, rcond=None)[0]
    
    L = coeffs[:Z_p.shape[0]]  # Past dependence
    G = coeffs[Z_p.shape[0]:]  # Future input dependence
    
    # Residuals
    residuals = Y_f.T - coeffs.T @ X
    
    return L, G, residuals

def wasserstein_radius(residuals, confidence=0.95):
    """Compute Wasserstein radius for given confidence level."""
    N = len(residuals)
    d = residuals.shape[1]
    # Concentration bound (simplified)
    epsilon = np.sqrt((2 * np.log(1/(1-confidence))) / N)
    return epsilon

def dr_mpc_step(L, G, residuals, u_future, z_current, 
                Q, R, epsilon, horizon):
    """Solve distributionally robust MPC step."""
    # Nominal prediction
    y_pred = L @ z_current + G @ u_future
    
    # Robust cost with worst-case expectation
    # Uses duality to convert to tractable form
    nominal_cost = (y_pred @ Q @ y_pred + u_future @ R @ u_future)
    
    # Robustification term (Wasserstein)
    robustness_penalty = epsilon * np.linalg.norm(residuals, axis=1).max()
    
    total_cost = nominal_cost + robustness_penalty
    
    return total_cost

Key Advantages

1. No model identification needed: Works directly from input-output data
2. Distribution-free: Makes no parametric assumptions on disturbances
3. Finite-sample guarantees: Performance bounds scale with √(1/N)
4. Tractable reformulation: DR problem reduces to convex optimization
5. Tunable conservatism: Wasserstein radius ε controls robustness/performance tradeoff

When to Use

• System dynamics unknown or poorly identified
• Disturbance distribution non-Gaussian or unknown
• Safety-critical applications requiring robustness guarantees
• Limited data but need statistical performance bounds
• Real-time control with computational constraints

Activation Keywords

• distributionally robust control
• data-driven predictive control
• Wasserstein MPC
• robust MPC unknown distribution
• subspace predictive control
• stochastic LTI control
• DR-DDPC