Robust MPC

Tube-based MPC is a foundational robust technique that guarantees the system's state remains within a bounded "tube" around a nominal trajectory. It decomposes the control law into two components:

• Nominal MPC: Computes a trajectory for a nominal, disturbance-free model.
• Ancillary Controller: A local feedback controller (e.g., linear state feedback) that rejects deviations, keeping the actual state within a tube around the nominal path. The tube's cross-section represents the robust positively invariant set, ensuring all possible uncertain trajectories are contained. This method explicitly handles bounded additive disturbances and is a cornerstone for many other robust MPC approaches.

Min-max MPC adopts a worst-case optimization philosophy. It solves an open-loop optimization problem that anticipates the most adversarial sequence of uncertainties or disturbances within a bounded set.

• Objective: Minimizes the maximum possible cost (the "worst-case" scenario) over the uncertainty set.
• Result: The computed control sequence is robust against all possible realizations but can be highly conservative.
• Challenge: The optimization is computationally intensive, often requiring a solution over all vertices of a polytopic uncertainty description. It is typically applied to systems with modest state dimensions due to this complexity.

Constraint tightening is a practical method to ensure robust constraint satisfaction by systematically shrinking the constraints used in the online optimization.

• Mechanism: The original state and input constraints are tightened (made more restrictive) by a margin that accounts for the worst-case effect of disturbances over the prediction horizon.
• Purpose: This creates a "backoff" from the true constraints, guaranteeing that when the actual, disturbed system evolves, it will not violate the original hard limits.
• Design: The tightening amount is derived from the size of the disturbance set and the system's dynamics. It is a key ingredient in tube-based and other robust MPC formulations to ensure recursive feasibility.

Stochastic MPC models uncertainties probabilistically, offering a less conservative alternative to worst-case methods. It uses chance constraints to manage risk.

• Chance Constraints: Specify that system constraints must be satisfied with a minimum probability (e.g., 95%), rather than always.
• Application: Ideal for problems with stochastic disturbances (like wind gusts) where occasional, small constraint violations are acceptable.
• Methods: Implementation often involves approximating chance constraints as deterministic tightened constraints or using scenario-based optimization. This technique is prominent in applications like renewable energy management and autonomous driving, where uncertainty is statistical in nature.

Multi-stage MPC, also known as feedback MPC or affine disturbance feedback, explicitly optimizes over closed-loop policies rather than open-loop control sequences. It models how future control actions can react to future uncertainty realizations.

• Policy Parameterization: Control inputs are expressed as affine functions of past disturbances: u_k = v_k + Σ Θ_{k,j} * w_j.
• Optimization Variables: The coefficients Θ of this feedback policy become decision variables in the optimization.
• Advantage: It is less conservative than open-loop min-max MPC because it accounts for the controller's ability to react in the future, leading to better performance while maintaining robustness. The computational complexity is higher than standard MPC but often lower than min-max.

Robust invariant sets are fundamental mathematical objects in the analysis and design of robust MPC, particularly for guaranteeing stability.

• Robust Positively Invariant (RPI) Set: A set of states such that if the system starts inside it, it remains inside for all future times despite bounded disturbances. Used to define the "tube" in tube-based MPC.
• Maximal Robust Invariant Set: The largest possible RPI set contained within the state constraints. Computing this set is often a prerequisite for robust MPC design to ensure long-term feasibility.
• Terminal Set: In robust MPC with a terminal constraint, this set is chosen to be robust invariant. This ensures that if the prediction horizon ends inside it, the system can be kept there indefinitely with a local controller, proving closed-loop robust stability.

Robust MPC is fundamental for the lateral and longitudinal control of self-driving cars. It computes steering and acceleration commands to follow a planned path while explicitly accounting for:

• Dynamic model uncertainties (e.g., tire friction, vehicle mass variation).
• External disturbances like crosswinds and road grade.
• Actuator limits (steering rate, brake pressure). The controller uses constraint tightening or tube-based methods to ensure the vehicle remains within a safe drivable corridor, even when predictions are imperfect, which is essential for functional safety (ISO 26262).

In continuous chemical manufacturing, Robust MPC manages reactors, distillation columns, and heat exchangers. Key applications include:

• Temperature and pressure regulation in exothermic reactors, where model errors can lead to runaway reactions.
• Product quality control by maintaining concentration within strict bounds despite feedstock variability.
• Economic optimization while respecting hard safety limits on vessel levels and flow rates. The robust invariant set concept ensures that all possible trajectories, given bounded parameter uncertainty, remain within the operating envelope, preventing costly shutdowns or hazardous conditions.

Robust MPC enables aggressive yet safe maneuvering for unmanned aerial vehicles (UAVs) and aircraft. It is used for:

• Precision landing in high wind shear, where aerodynamic coefficients are uncertain.
• Formation flying of drone swarms, requiring robust collision avoidance under communication delays.
• Re-entry vehicle guidance, managing extreme aerodynamic and thermal uncertainties. Techniques like min-max MPC or scenario-based MPC are employed to handle the worst-case disturbance within known bounds, ensuring the vehicle never violates structural or thermal constraints.

For robotic arms interacting with uncertain environments, Robust MPC is critical for force-controlled tasks and compliant manipulation. It addresses:

• Unknown object properties (mass, inertia, friction) during pick-and-place.
• Unmodeled contact dynamics when pushing or inserting parts.
• Joint flexibility and gearbox backlash in heavy-duty manipulators. By incorporating disturbance observers and tightening contact force constraints, the controller can execute delicate assembly tasks or human-robot collaboration without causing damage or excessive force.

Robust MPC optimizes energy flows in systems with volatile supply and demand. Primary use cases are:

• Microgrid dispatch: Coordinating renewable sources (solar, wind), batteries, and diesel generators despite forecast errors in generation and load.
• Building HVAC control: Maintaining temperature comfort zones with uncertain occupancy and weather, while minimizing energy cost.
• Battery state-of-charge management in electric vehicles, accounting for aging effects and temperature-dependent efficiency. Chance-constrained MPC, a stochastic variant of robust MPC, is often used here to balance constraint violations with economic performance probabilistically.

In semiconductor manufacturing and advanced CNC machining, Robust MPC drives stages and tools with nanometer precision. It compensates for:

• Parasitic dynamics from cabling and cooling lines.
• Non-linear friction (Stribeck effect) and ripple forces in linear motors.
• Vibration modes that are difficult to model accurately. The controller uses a nominal model for performance but optimizes over a tube of trajectories to ensure the actual position error always stays within the tolerance budget, guaranteeing yield in photolithography and high-speed milling.

Tube-Based MPC is a dominant method in Robust MPC that guarantees constraint satisfaction by maintaining the system's state within a bounded 'tube' around a nominal, disturbance-free trajectory. The controller consists of two components:

• A nominal MPC that plans a trajectory for the idealized model.
• An ancillary feedback controller (often a simple linear controller) that actively rejects disturbances, keeping the real system's state within a predefined cross-section (the tube) around the nominal path. Constraint sets are recursively tightened offline to ensure the entire tube remains feasible.

Min-Max MPC adopts a worst-case optimization philosophy. It solves an open-loop optimization problem where the control sequence is computed to minimize the maximum possible cost (the 'worst-case' scenario) across all admissible realizations of uncertainty within a bounded set. This leads to a conservative but safe control policy. The optimization is computationally challenging as it involves solving a minimax problem, often requiring scenario-based or constraint sampling approximations.

Constraint tightening is a fundamental technique in Robust MPC to ensure recursive feasibility. The key idea is to preemptively shrink the constraint sets used in the online optimization to account for the effect of future disturbances. The amount of tightening is computed offline based on the assumed bounds of the uncertainty. This creates a 'robust positive invariant' set. As the prediction horizon extends, constraints are tightened more aggressively, ensuring that all possible future state trajectories, given the uncertainty, will satisfy the original, untightened constraints.

A Robust Positive Invariant (RPI) Set is a core concept in stability analysis for Robust MPC. It is a set of states such that, if the system state enters it, it will remain inside for all future time steps for any admissible sequence of disturbances, given a specific control law. In Tube-Based MPC, the 'tube' cross-section is often designed as an RPI set. The terminal constraint in a robust MPC formulation is frequently chosen to be an RPI set to guarantee persistent constraint satisfaction and stability.

Linear Matrix Inequalities (LMIs) are a powerful computational tool for the offline design of Robust MPC controllers, particularly for linear systems with polytopic or norm-bounded uncertainty. Stability conditions, invariant set computations, and feedback gain design for the ancillary controller in tube-based approaches can often be formulated as LMI feasibility or optimization problems. Solving these LMIs provides the pre-computed parameters (like constraint tightening margins or feedback gains) that ensure robust performance online, shifting computational burden from the real-time loop to the design phase.