mpc-plant-model-mismatch

name: mpc-plant-model-mismatch description: "Model Predictive Control stability analysis under plant-model mismatch. Covers discounted/infinite-horizon optimal control, suboptimality bounds, and robustness guarantees. Use for: MPC design, control system robustness analysis, plant-model mismatch tolerance, stability proofs, optimal control with surrogate models." category: control-theory

MPC Stability under Plant-Model Mismatch

Description

Unified framework for analyzing MPC (Model Predictive Control) and infinite-horizon optimal control stability when using surrogate models. Provides:

• Exponential stability guarantees under proportional mismatch bounds
• Suboptimality bounds for closed-loop cost
• Uniform robustness across horizon lengths (longer horizons don't require smaller mismatch tolerance)
• Practical design checklist for MPC engineers

Activation Keywords

• MPC plant-model mismatch
• model predictive control robustness
• discounted MPC
• infinite-horizon optimal control
• stability analysis
• surrogate model control
• 模型预测控制
• 模型不匹配
• 鲁棒控制

Tools Used

• read: Read control theory papers and derivations
• write: Generate MPC implementation code
• exec: Run simulation to verify stability bounds
• web_search: Find related control theory references

Instructions for Agents

When a user asks about MPC robustness under plant-model mismatch:

1. Problem setup: Help the user define the real plant vs. surrogate model mismatch bounds
2. Check stability conditions: Verify continuity, cost-controllability, and mismatch magnitude
3. Apply the checklist: Walk through the MPC design checklist
4. Calculate bounds: Compute suboptimality bounds based on mismatch magnitude
5. Parameter selection: Guide on horizon length and discount factor selection
6. Reference theory: Point to the theorems and proofs for deeper understanding

Examples

User: Does MPC stability hold when my model has plant-model mismatch?
Agent: Using the MPC Plant-Model Mismatch framework, we can analyze this. If your mismatch is bounded by proportional bounds and the cost-controllability condition holds, exponential stability is guaranteed even for longer prediction horizons...

Core Theory

Problem Setup

• Real plant: Unknown dynamics x+ = f(x,u) (continuous)
• Surrogate model: Approximate dynamics x+ = g(x,u) (continuous)
• Plant-model mismatch: |f(x,u) - g(x,u)| ≤ γ|x| + δ|u| (proportional bounds)
• Assumption: Origin remains equilibrium under mismatch
• Cost: Quadratic l(x,u) = x'Qx + u'Ru (positive definite)

Stability Guarantee

Theorem: Under continuity + cost-controllability, exponential stability holds if:

γ, δ sufficiently small (proportional mismatch bounds exist)

Key insight: Stability guarantee is uniform over horizon length — longer horizons don't require smaller mismatch tolerance.

Suboptimality Bound

Closed-loop cost: J_cl = Σ l(x_k, u_k) (real plant trajectory) Surrogate optimal cost: V_opt^g (computed on surrogate model)

Bound: J_cl ≤ V_opt^g + ε(γ, δ) where ε depends on mismatch magnitude.

Tradeoff Relationship

Horizon N  ↑ → Better approximation, but computation ↑
Discount λ ↓ → Tighter stability, but mismatch tolerance ↓
Mismatch γ ↓ → Better stability, but requires better model

The framework reveals how these three parameters interact.

Key Results

1. Infinite-Horizon Stability

For discounted infinite-horizon MPC:

• Stability preserved under plant-model mismatch
• Suboptimality bound scales with mismatch magnitude
• Discount factor λ affects robustness margin

2. Finite-Horizon Stability

For standard finite-horizon MPC:

• Same stability guarantees hold
• Horizon length doesn't tighten mismatch requirements
• Practical: Can use longer horizons without stricter model accuracy

3. Cost-Controllability Assumption

Essential condition: Surrogate model satisfies cost-controllability

Exists α > 0 such that V_g(x) ≤ α|x|²  (value function bounded)

Practical Applications

Control System Design

1. Model tolerance specification: Determine acceptable plant-model mismatch bounds
2. Horizon selection: Choose horizon length freely (no mismatch penalty)
3. Discount tuning: Balance robustness vs. performance

Robustness Analysis Workflow

# Step 1: Estimate plant-model mismatch
γ_model = estimate_model_error()  # From data/physics
δ_control = estimate_control_error()

# Step 2: Check stability condition
if γ_model < γ_threshold and δ_control < δ_threshold:
    # Stability guaranteed
    pass

# Step 3: Compute suboptimality bound
epsilon = compute_suboptimality(γ_model, δ_control)

Real-World Scenarios

• Process control: Plant dynamics estimated from data (inevitable mismatch)
• Robotics: Physics model approximates real dynamics
• Power systems: Grid model vs. actual load/generation
• Aerospace: Flight dynamics model vs. airframe reality

Mathematical Framework

Quadratic Cost Setup

Stage cost: l(x,u) = x'Qx + u'Ru  (Q,R > 0)
Value function: V(x) = min Σ λ^k l(x_k,u_k)
Terminal cost: V_f(x) bounds tail cost

Lyapunov Stability Proof

Key idea: Use value function as Lyapunov candidate

V(x_k+1) - V(x_k) ≤ -l(x_k,u_k) + perturbation_from_mismatch

Stability if perturbation bounded by stage cost decrease.

Cost-Controllability Condition

Surrogate model property:

V_g(x) ≤ α|x|²  (value function quadratic bound)

This ensures stability analysis valid.

Implementation Guidance

MPC Design Checklist

Model Improvement Strategy

If mismatch too large for stability:

1. Refine model: Better physics understanding, more data
2. Adaptive MPC: Online model correction
3. Robust MPC: Explicit uncertainty handling (tube MPC)
4. Discount adjustment: Lower λ for tighter robustness

Key Insights

1. Uniform robustness: Horizon length doesn't tighten mismatch requirements — counterintuitive but powerful
2. Unified framework: Single theory covers finite/infinite horizon, discounted/undiscounted cases
3. Practical tolerance: Explicit mismatch bounds enable model validation
4. Tradeoff understanding: Horizon/discount/mismatch interaction guides design choices

Related Concepts

• Tube MPC: Explicit robustness via constraint tightening
• Adaptive MPC: Online model learning
• Gain scheduling: Multiple models for different operating points
• Robust control: Worst-case design (H∞, μ-synthesis)

References

arXiv: 2604.08521v1
Authors: Moldenhauer, Worthmann, Postoyan, Nešić, Granzotto
Categories: math.OC, eess.SY
Title: "Discounted MPC and infinite-horizon optimal control under plant-model mismatch: Stability and suboptimality"