Stability (Nominal and Robust)

Nominal stability guarantees that the closed-loop system converges to its target (e.g., a setpoint or reference trajectory) when the controller's internal model is a perfect representation of the real plant. It is the baseline assumption. Key design methods to ensure nominal MPC stability include:

• Terminal Cost: Adding a final state penalty (often a Lyapunov function) to the objective.
• Terminal Constraint: Requiring the predicted state at the end of the horizon to lie within a positively invariant terminal set.
• Infinite Horizon Approximation: Using a sufficiently long prediction horizon to approximate an infinite-horizon optimal control problem, whose solution is inherently stabilizing.

Robust stability ensures the closed-loop system remains stable despite the presence of model mismatch, external disturbances, or measurement noise. Since a perfect model is impossible in practice, robust stability is essential for real-world deployment. Common MPC strategies for robustness include:

• Tube-Based MPC: Maintains the actual system state within a bounded 'tube' around the nominal predicted trajectory. Control actions are computed to keep this tube inside constraints.
• Min-Max (Open-Loop Feedback) MPC: Optimizes control inputs against the worst-case realization of uncertainty within a bounded set.
• Constraint Tightening: Artificially tightens the constraints used in the online optimization to create a robust positively invariant set, ensuring the real system never violates the true physical limits.

The terminal set is a critical concept for proving MPC stability. It is a region of the state space, often designed around the target equilibrium, with two key properties:

• Positive Invariance: If the state enters this set, it will remain inside under a local stabilizing controller (e.g., a Linear Quadratic Regulator).
• Constraint Satisfaction: All states within the terminal set satisfy the system's operational constraints. In MPC design, a terminal constraint forces the predicted state at the end of the horizon to lie within this set. This allows the finite-horizon MPC problem to inherit the stability guarantees of the infinite-horizon problem, as the local controller can take over safely beyond the prediction window.

A Lyapunov function, (V(x)), is a scalar, energy-like function used to mathematically prove stability. For a stable equilibrium (e.g., (x=0)), a valid Lyapunov function must be:

• Positive Definite: (V(x) > 0) for all (x \neq 0), and (V(0)=0).
• Decreasing: Its value must decrease over time, i.e., (\Delta V(x) = V(x_{k+1}) - V(x_k) < 0). In MPC, the optimal cost function (J^(x_k)) (the value of the solved optimization problem at time step (k)) is often used as a Lyapunov function candidate. Demonstrating that (J^(x_{k+1}) - J^*(x_k) \leq 0) proves the system's state is moving toward lower 'cost energy,' guaranteeing convergence and stability.

Recursive feasibility is a prerequisite for stability. It guarantees that if an optimization problem is feasible (a solution exists) at the initial time step, it will remain feasible at every subsequent time step. If feasibility is lost, the MPC controller fails, potentially leading to instability. Recursive feasibility is ensured through design features like:

• Proper design of the terminal set and terminal constraint.
• The use of soft constraints (with penalty terms) for output constraints, ensuring a solution always exists, though potentially at a higher cost.
• Constraint tightening in robust MPC, which proactively reduces the feasible region to account for future uncertainty, ensuring all future predicted states remain within the true constraints.

There is a fundamental trade-off between stability guarantees and control performance.

• Aggressive, high-performance controllers (e.g., with short horizons, high gain) may provide excellent tracking and fast response but risk instability if disturbances occur or models are inaccurate.
• Conservative, robustly stable controllers (e.g., with long horizons, constraint tightening) guarantee stability under a wider range of conditions but may exhibit slower, less responsive performance. The MPC designer must balance this trade-off. Techniques like robust MPC explicitly manage this by characterizing uncertainty sets: larger uncertainty sets lead to more conservative (but more robust) control actions, while smaller sets allow for more aggressive performance.

A Lyapunov function is a scalar, energy-like function used to prove the asymptotic stability of a system's equilibrium point. In MPC stability analysis, it provides a formal certificate of convergence.

• Role in MPC: A common design approach is to choose a terminal cost that is a Lyapunov function for the system under a local controller.
• Decreasing Property: Stability is proven by showing this function's value decreases along the system's trajectory.
• Example: For a linear system, a quadratic function (V(x) = x^T P x) (where (P) is a positive definite matrix) is often used, directly connecting to Linear Quadratic Regulator (LQR) theory.

The terminal cost and terminal constraint are the primary design tools used to enforce nominal stability in MPC. They effectively turn a finite-horizon optimization into an approximation of an infinite-horizon problem.

• Terminal Cost: An additional term, (V_f(x(N))), added to the MPC cost function, penalizing the state at the end of the prediction horizon. It is often chosen as a Lyapunov function.
• Terminal Constraint: A requirement that the predicted state at the horizon's end, (x(N)), must lie within a terminal set (\mathbb{X}_f). This set is designed to be positively invariant under a local stabilizing controller.
• Combined Guarantee: Using both ensures the optimized trajectory drives the system to a region where a simple backup controller can maintain stability, guaranteeing closed-loop convergence.

Robust MPC is a family of MPC strategies that guarantee robust stability and robust constraint satisfaction despite bounded model uncertainty, disturbances, or noise.

• Core Challenge: A standard MPC controller using a nominal model can become unstable if the real plant deviates from that model.
• Key Techniques:
  • Tube-based MPC: Optimizes a trajectory for the nominal model while maintaining the real state within a bounded "tube" around it using an ancillary feedback controller.
  • Constraint Tightening: The constraints in the online optimization are conservatively tightened to account for worst-case uncertainty, ensuring the real system never violates the original constraints.
  • Min-Max MPC: Solves a worst-case optimization over a set of possible models or disturbances, which is computationally intensive.
• Outcome: The controller maintains performance and safety guarantees across a defined spectrum of model error.

Stochastic MPC addresses uncertainty by modeling disturbances or parameters probabilistically. Instead of worst-case guarantees, it enforces chance constraints, which must be satisfied with a specified probability.

• Philosophy: Less conservative than Robust MPC, as it does not plan for the absolute worst-case scenario, which may be very unlikely.
• Chance Constraints: Constraints of the form (\mathbb{P}( g(x,u) \leq 0 ) \geq 1 - \alpha ), where (\alpha) is a small risk parameter (e.g., 0.05).
• Stability: Stability is typically defined in a probabilistic sense (e.g., mean-square stability). Guarantees often rely on assumptions about disturbance distributions and may use recursive feasibility in expectation.
• Application: Ideal for systems with well-characterized stochastic noise, such as financial portfolios, energy management with renewable forecasts, or process control.

Recursive feasibility is a fundamental property required for any stability guarantee in MPC. It ensures that if an optimization problem is feasible (a solution exists) at the current time step, it will remain feasible at all future time steps.

• Critical Importance: Loss of feasibility means the solver cannot find a control sequence that satisfies all constraints, leading to undefined behavior and potential system failure.
• How it's Ensured:
  • Proper design of terminal constraints is the standard method for nominal MPC.
  • In Robust MPC, techniques like constraint tightening and tube-based approaches are used to maintain recursive feasibility despite uncertainty.
• Consequence: Without recursive feasibility, formal stability proofs (which rely on the continued existence of a Lyapunov function) break down.

Input-to-State Stability (ISS) is a stability framework for nonlinear systems subject to disturbances. It quantifies how the system's state magnitude depends on the magnitude of the disturbance input.

• Relevance to Robust MPC: It provides a rigorous, nonlinear extension of stability analysis for systems with persistent disturbances. A controller achieving ISS guarantees that the state will remain bounded within a ball whose size is proportional to the disturbance bound.
• ISS-Lyapunov Function: Stability is proven using an ISS-Lyapunov function, which must satisfy a stronger condition than a standard Lyapunov function, explicitly accounting for the disturbance.
• Practical Interpretation: Even if disturbances prevent convergence to zero, ISS ensures the system's deviation is predictable and proportional to the disturbance strength. This is a crucial analysis tool for Robust MPC and Nonlinear MPC (NMPC) in practical, noisy environments.