MPPI Control: Theory and Practice

Introduction

Model Predictive Path Integral (MPPI) control represents a paradigm shift from traditional control strategies. Unlike PID controllers that react to current errors, or conventional MPC that solves optimization problems at each timestep, MPPI uses stochastic sampling and information-theoretic principles to find optimal control trajectories.

For chemical reactors with nonlinear dynamics, multiple interacting variables, and complex constraints, MPPI offers significant advantages over traditional approaches.

Theoretical Foundation

From Optimal Control to Path Integrals

Traditional optimal control seeks to minimize a cost function over a control horizon:

MPPI reformulates this problem using concepts from statistical mechanics. Instead of solving a deterministic optimization, it considers the probability distribution over trajectories and uses importance sampling to estimate the optimal control.

Information-Theoretic Interpretation

The key insight of MPPI is that optimal control can be framed as minimizing the KL-divergence between the controlled trajectory distribution and an uncontrolled (prior) distribution, subject to a cost constraint:

Temperature Parameter

The "temperature" parameter λ controls the trade-off between exploration and exploitation:

• High λ: More exploration, smoother control, robust to noise
• Low λ: More aggressive optimization, faster convergence to optimal

The MPPI Algorithm

The core MPPI algorithm proceeds as follows:

Pseudocode

def mppi_control(state, model, cost_fn, K, T, λ):
    # Initialize control sequence
    U = initial_control_sequence(T)

    for iteration in range(max_iterations):
        # Sample K perturbations
        noise = sample_gaussian(K, T, Σ)

        # Roll out trajectories
        costs = []
        for k in range(K):
            trajectory = simulate(state, U + noise[k], model)
            costs.append(cost_fn(trajectory))

        # Compute weights
        β = min(costs)
        weights = exp(-(costs - β) / λ)
        weights = weights / sum(weights)

        # Update control sequence
        U = U + sum(weights * noise)

    return U[0]  # Return first action

Advantages for Process Control

Practical Implementation

Model Requirements

MPPI requires a forward model that can predict process evolution. In Acaysia, we typically use hybrid gray-box models that combine:

• First-principles equations (mass balances, energy balances)
• Neural network components for unknown kinetics or heat transfer
• Online adaptation for model correction

Cost Function Design

The cost function encodes all control objectives. A typical structure for reactor control:

def reactor_cost(trajectory, target):
    cost = 0

    # Tracking cost (yield optimization)
    cost += w_track * ||T - T_target||²

    # Control effort (energy efficiency)
    cost += w_effort * ||Δu||²

    # Constraint penalties (safety)
    cost += w_temp * relu(T - T_max)²
    cost += w_press * relu(P - P_max)²

    # Terminal cost (batch completion)
    cost += w_terminal * ||x_final - x_target||²

    return cost

Computational Considerations

Tuning Guidelines

Temperature Selection

Start with λ = 1 and adjust based on observed behavior:

• If control is too aggressive or oscillatory → increase λ
• If control is too slow or conservative → decrease λ
• If system doesn't reach setpoints → check model accuracy first

Sample Count

More samples improve solution quality but increase computation time. Guidelines:

• Start with K = 1000 for initial testing
• Increase until performance improvement plateaus
• Use GPU acceleration for K > 2000

Cost Function Weights

Case Example: Batch Reactor Temperature Control

Consider a batch reactor with an exothermic reaction. The goal is to follow a temperature profile while respecting safety constraints.

Challenge

Traditional PID struggles because:

• Reaction rate is highly temperature-dependent (Arrhenius kinetics)
• Heat release varies with conversion
• Cooling capacity is limited
• Need to anticipate exothermic runaway

MPPI Solution

MPPI naturally handles this scenario by:

1. Predicting temperature evolution over the control horizon
2. Evaluating thousands of cooling trajectories
3. Selecting trajectories that track the profile while avoiding runaway
4. Proactively increasing cooling before exothermic peaks

Results

In controlled testing, MPPI achieved:

• 40% reduction in temperature deviation from setpoint
• 25% faster batch completion times
• Zero safety limit excursions
• 15% reduction in cooling utility consumption