mpc-stability-suboptimality

name: mpc-stability-suboptimality description: Model Predictive Control (MPC) stability and suboptimality analysis under plant-model mismatch. Covers discounted and undiscounted infinite-horizon optimal control, stability guarantees with model uncertainty, and suboptimality bounds. Use when analyzing MPC robustness, handling model-plant mismatch in control systems, or implementing robust MPC controllers.

MPC Stability and Suboptimality Under Plant-Model Mismatch

Overview

This skill provides theoretical foundations and practical guidance for implementing robust Model Predictive Control (MPC) when the model differs from the real plant. It covers stability guarantees, suboptimality bounds, and the tradeoff between horizon length, discounting, and model mismatch.

Key Concepts

Plant-Model Mismatch

When the surrogate model used for MPC differs from the actual plant dynamics:

Model uncertainty bounds: Proportional to states and controls
Equilibrium preservation: Origin remains an equilibrium under mismatch
Continuity requirements: Model and cost-controllability assumptions

Discounted vs. Undiscounted Scenarios

Discounted Control:

Running cost: γ^k * q(x_k, u_k) with discount factor γ ∈ (0, 1)
Infinite-horizon cost: finite even with undiscounted stage cost
Less conservative for long horizons

Undiscounted Control:

Running cost: q(x_k, u_k) without discounting
Requires stability for finite infinite-horizon cost
More sensitive to horizon length

Stability Guarantees

Exponential Stability Conditions:

Model continuity at equilibrium
Cost-controllability property
Plant-model mismatch bounds
Uniform guarantees over horizon length

Key Insight: Larger horizons do NOT require successively smaller mismatch bounds - robustness is uniform.

Suboptimality Analysis

Closed-Loop Cost Bound:

Recovers optimal cost of surrogate model
Tradeoff: Horizon length vs. Discount factor vs. Mismatch size
Quantifies performance loss from model error

Framework

Unified Quadratic Cost Framework

# General MPC formulation
def mpc_cost(x_sequence, u_sequence, model, gamma):
    total_cost = 0
    for k in range(horizon):
        stage_cost = q(x_sequence[k], u_sequence[k])
        total_cost += (gamma ** k) * stage_cost
    
    # Terminal cost for stability
    if terminal_constraint:
        total_cost += terminal_cost(x_sequence[-1])
    
    return total_cost

Plant-Model Mismatch Assumption

||f_plant(x, u) - f_model(x, u)|| ≤ α * ||x|| + β * ||u||

where:
- f_plant: actual plant dynamics
- f_model: surrogate model dynamics
- α, β: mismatch bounds (proportional to states and controls)

Stability Analysis Procedure

Check continuity: Model continuous at equilibrium
Verify cost-controllability: Can drive cost to zero from any state
Compute mismatch bounds: α and β parameters
Select horizon: Tradeoff with computational limits
Set discount factor: If using discounted formulation

Design Tradeoffs

Horizon Length vs. Mismatch Bounds

Tradeoff:

Longer horizon: Better optimality, same stability robustness
Mismatch bounds: Uniform stability guarantees
Insight: Robustness doesn't degrade with longer horizons

Discount Factor Effects

Discount factor γ affects:
- Cost finiteness: γ < 1 ensures finite infinite-horizon cost
- Conservatism: Lower γ reduces sensitivity to distant future
- Tradeoff: γ vs. horizon length for optimal performance

Suboptimality vs. Stability

Suboptimality Bound:

Quantifies deviation from surrogate optimal cost
Depends on mismatch size and horizon/discount choices
Provides performance guarantee under uncertainty

Practical Implementation

Robust MPC Design Steps

Model Selection:
- Choose surrogate model close to plant behavior
- Estimate mismatch bounds from data
- Ensure equilibrium preservation
Cost Function Design:
- Quadratic costs: Easy to analyze
- Terminal cost: For stability guarantees
- Stage cost: Reflect true objectives
Horizon and Discount Selection:
- Horizon: Balance computation vs. optimality
- Discount: Consider long-term vs. short-term goals
- Tradeoff: Use analysis to guide selection
Implementation:
- Online optimization at each timestep
- Handle constraints explicitly
- Warm-start from previous solution

Implementation Code Pattern

class RobustMPCController:
    def __init__(self, model, horizon, gamma, mismatch_bounds):
        self.model = model
        self.N = horizon
        self.gamma = gamma
        self.alpha, self.beta = mismatch_bounds
        
    def solve(self, current_state):
        # Solve finite-horizon optimization
        optimal_sequence = self.optimize(current_state)
        
        # Apply first input
        u0 = optimal_sequence[0]
        
        # Stability check (optional)
        if self.check_stability(current_state, u0):
            return u0
        else:
            return self.fallback_action(current_state)
    
    def optimize(self, x0):
        # Formulate and solve optimization problem
        # minimize: sum_{k=0}^{N-1} γ^k * q(x_k, u_k)
        # subject to: x_{k+1} = f_model(x_k, u_k)
        return self.solve_qp(x0)

Applications

Process Control

Chemical reactors with uncertain kinetics
Heat exchangers with varying parameters
Batch processes with model drift

Autonomous Systems

Vehicle control with uncertain dynamics
Drone navigation with wind uncertainty
Robot manipulation with load changes

Energy Systems

Power grid control with demand uncertainty
Building HVAC with thermal model mismatch
Battery management with degradation uncertainty

Mathematical Results

Exponential Stability Theorem

Under plant-model mismatch bounds and cost-controllability:

Closed-loop origin is exponentially stable
Rate depends on model properties and mismatch bounds
Guarantee uniform over horizon length N

Suboptimality Bound

Closed-loop cost satisfies:

J_closed_loop ≤ J_optimal_surrogate * (1 + ε(mismatch, horizon, gamma))

where ε quantifies suboptimality from model error

Key Lemma: Uniform Robustness

For any horizon N, stability guarantee depends only on:

Mismatch bounds (α, β)
Cost-controllability constant
Model continuity

NOT on horizon length.

References

Paper: arxiv:2604.08521v1
PDF: https://arxiv.org/pdf/2604.08521v1
Authors: Robert H. Moldenhauer, Karl Worthmann, Romain Postoyan, Dragan Nešić, Mathieu Granzotto
Categories: math.OC, eess.SY
Published: 2026-04-09

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