discounted-mpc-control

name	discounted-mpc-control
description	Model Predictive Control (MPC) stability and suboptimality analysis under plant-model mismatch with discounting. Provides theoretical guarantees for infinite-horizon optimal control when using surrogate models. Use when designing robust control systems, analyzing MPC stability, or dealing with model uncertainty in control applications.

Discounted MPC and Infinite-Horizon Optimal Control Under Plant-Model Mismatch

Core Problem

Real-world control systems operate using models that differ from the actual plant:

• Parameter uncertainties
• Unmodeled dynamics
• Linearization errors
• Time-varying system behavior

This plant-model mismatch affects stability and performance of Model Predictive Control (MPC).

Key Contributions

1. Unified Framework

Analysis covers both:

• Finite-horizon MPC: Standard receding horizon control
• Infinite-horizon optimal control: Value iteration / policy optimization

Both discounted and undiscounted scenarios included.

2. Stability Guarantees

Theorem: Under plant-model mismatch bounds, exponential stability is guaranteed if:

• Model continuity holds
• Cost-controllability is satisfied
• Origin remains equilibrium under mismatch

3. Suboptimality Bounds

When using surrogate model with mismatch bounds proportional to states and controls:

J_actual(x0) - J_optimal(x0) ≤ γ * ||mismatch||

Where:

• J_actual: Closed-loop cost with real plant
• J_optimal: Optimal cost for surrogate model
• γ: Derived constant based on problem structure

4. Trade-off Analysis

Key insight: Robustness guarantees are uniform over horizon length

Larger prediction horizons do NOT require successively smaller plant-model mismatch for stability.

Mathematical Framework

Plant-Model Mismatch Model

f_actual(x, u) = f_model(x, u) + Δ(x, u)

Where Δ(x, u) represents mismatch bounded by:

||Δ(x, u)|| ≤ α||x|| + β||u||

Discounted Cost Function

V(x) = Σ_{k=0}^∞ γ^k * l(x_k, u_k)

Discount factor γ ∈ (0, 1) affects:

• Convergence rate
• Suboptimality gap
• Required mismatch tolerance

Practical Implications

For Control System Design

1. Tolerance specification: Determine acceptable mismatch levels
2. Horizon selection: Longer horizons don't hurt robustness
3. Discount tuning: Balance performance vs robustness

For System Identification

1. Identify critical model parameters
2. Quantify acceptable model error
3. Guide data collection priorities

Design Guidelines

When to Use Discounted MPC

• Long planning horizons
• Significant model uncertainty
• Need for robust stability guarantees
• Safety-critical applications

Parameter Selection

Parameter	Effect	Trade-off
Horizon N	Better optimality	Computation
Discount γ	Robustness	Performance
Mismatch bound	Stability	Model complexity

Related Control Techniques

• Robust MPC
• Adaptive control
• Tube-based MPC
• Data-driven control

Paper Reference

Title: Discounted MPC and infinite-horizon optimal control under plant-model mismatch: Stability and suboptimality Authors: Robert H. Moldenhauer, Karl Worthmann, Romain Postoyan, Dragan Nešić, Mathieu Granzotto arXiv: 2604.08521 Category: math.OC, eess.SY Published: 2026-04-09 Submitted to: 65th IEEE Conference on Decision and Control

Key Equations

Stability Condition

V(x_{k+1}) - V(x_k) ≤ -α||x_k||² + β||Δ||²

Suboptimality Bound

V_closed_loop ≤ V_optimal * (1 + κ)

Where κ depends on mismatch magnitude and discount factor.