Model Predictive Control (MPC)

MPC uses an internal dynamic model of the system to simulate and predict its future behavior over a defined, finite time window called the prediction horizon. This forward-looking capability allows the controller to anticipate the consequences of its actions, such as future obstacle positions or system inertia, enabling proactive rather than purely reactive control. For example, an autonomous vehicle using MPC can predict its trajectory several seconds ahead to smoothly navigate a curve.

At each control step, MPC solves a real-time optimization problem to find the best sequence of future control inputs. Crucially, this optimization explicitly handles hard constraints and soft constraints on:

• State Variables: e.g., joint angles, velocity limits.
• Control Inputs: e.g., actuator torque, steering angle limits.
• Outputs: e.g., staying within a safe corridor. This allows the controller to operate safely at the very limits of performance, such as a robot arm moving at maximum speed without exceeding its torque limits.

MPC implements a receding horizon or moving horizon strategy. After solving the optimization, only the first control input from the computed optimal sequence is applied to the system. At the next time step, the horizon "recedes" forward, new sensor measurements are incorporated to update the state estimate, and the optimization is solved again. This feedback mechanism provides inherent robustness to model inaccuracies and external disturbances, as the plan is constantly corrected based on real-world observations.

The performance of MPC is fundamentally tied to the accuracy of its internal model. This model, which can be linear, non-linear, or hybrid, mathematically describes the system's dynamics—how its state evolves in response to control inputs. Common forms include:

• Linear Time-Invariant (LTI) Models: Enable fast Quadratic Programming (QP) solves.
• Non-Linear Models: Provide higher fidelity for complex systems like agile drones, but require more computationally intensive Non-Linear Programming (NLP) solvers. The model is the controller's "world simulator," and its fidelity dictates how well predictions match reality.

The optimization is guided by a cost function (or objective function) that the controller minimizes. This function encodes competing control goals as weighted terms, allowing for sophisticated trade-offs. A typical cost function for a mobile robot might sum:

• Tracking Error: Deviation from a desired path.
• Control Effort: Minimization of energy use or actuator wear.
• Smoothness: Penalty for jerky control inputs.
• Terminal Cost: A penalty to ensure the predicted state at the end of the horizon is stable. Engineers tune these weights to define the desired "personality" of the controller.

The need to solve an optimization problem online, within one control cycle, is MPC's primary engineering challenge. This demands:

• Specialized Solvers: Fast, reliable numerical optimization algorithms (e.g., active-set, interior-point).
• High-Performance Hardware: Often deployed on dedicated computing units (GPUs, FPGAs) for complex non-linear problems.
• Model Simplification: Using linearized or reduced-order models to meet timing deadlines. The trade-off between model complexity, prediction horizon length, and available compute power is a central design consideration for real-time robotic systems.

MPC is the dominant advanced control technology in refineries, chemical plants, and pharmaceutical manufacturing. It manages complex, interacting variables like temperature, pressure, and flow rates in distillation columns, reactors, and heat exchangers.

• Core Function: Maximizes yield, minimizes energy consumption, and ensures safe operation within strict equipment constraints.
• Example: Controlling a crude oil distillation unit to optimize the production of gasoline, diesel, and other fractions while respecting column flooding and temperature limits.
• Economic Impact: Typical benefits include a 2-5% increase in throughput and a 10-20% reduction in energy costs.

In autonomous driving, MPC is used for path tracking and trajectory planning. It solves the problem of following a desired path while satisfying vehicle dynamics, actuator limits, and safety constraints.

• Core Function: Computes optimal steering, throttle, and brake inputs to smoothly and safely follow a planned trajectory, anticipating curves and obstacles.
• Key Advantage: Explicitly handles constraints like steering angle limits, tire friction circles, and collision avoidance boundaries.
• Integration: Works alongside perception and high-level planning modules, executing smooth, stable low-level control.

MPC is employed for attitude control, trajectory optimization, and energy management in aircraft, spacecraft, and drones. It is particularly valuable for systems with complex dynamics and stringent safety requirements.

• Core Function: Manages control surfaces (ailerons, elevators, rudders) or thrusters to achieve precise maneuvers while minimizing fuel consumption and respecting structural load limits.

• Example Applications:

  • Quadrotor Drones: Stabilizing flight and executing agile maneuvers.
  • Re-entry Vehicles: Managing aerodynamic heating and g-forces during atmospheric descent.
  • Satellites: Performing fuel-efficient orbital station-keeping and attitude adjustments.

For robotic arms and legged robots, MPC enables dynamic, torque-controlled motion that is responsive to disturbances and adaptable to changing tasks. It is central to modern, high-performance robotic control.

• Core Function: Computes optimal joint torques or motor currents to achieve desired end-effector forces or body motions, considering full-body dynamics and contact forces.
• Key Use Cases:
  • Dynamic Walking & Running: For humanoids and quadrupeds like Boston Dynamics' Atlas or Spot.
  • Force-Controlled Manipulation: Performing assembly tasks or handling fragile objects.
  • Payload Transportation: A mobile manipulator moving a heavy, sloshing liquid.
• Challenge: Requires solving the optimization problem at high frequencies (100-1000 Hz), demanding efficient solvers and models.

MPC optimizes the operation of renewable energy systems, building HVAC, and smart grids. It balances supply and demand, manages energy storage, and improves efficiency in the face of uncertainty and fluctuating prices.

• Core Function: Schedules energy generation, storage, and consumption over a future horizon to minimize cost or maximize renewable utilization.
• Example Applications:
  • Solar-Thermal Power Plants: Pre-heating systems based on weather forecasts to maintain output during cloud cover.
  • Building Management: Pre-cooling a building at night using cheaper electricity to reduce daytime HVAC load.
  • Microgrids: Coordinating solar panels, batteries, and diesel generators to ensure stable power.

MPC provides precise, adaptive control for medical devices like artificial pancreases, powered prosthetics, and exoskeletons. It personalizes treatment and assistance based on physiological models and real-time sensor data.

• Core Function: Adjusts device outputs (e.g., insulin delivery, joint torque) to track a desired physiological state (e.g., blood glucose, limb position) while ensuring patient safety.
• Key Applications:
  • Artificial Pancreas: Automatically regulates insulin and glucagon delivery for people with type 1 diabetes, using a model of glucose metabolism.
  • Robotic Exoskeletons: Assists human walking or rehabilitation, adapting support based on the user's gait and intent.
• Critical Feature: The ability to enforce hard constraints (e.g., maximum safe insulin dose) is essential for patient safety.

PID control is a ubiquitous linear feedback control algorithm that calculates an error value as the difference between a desired setpoint and a measured process variable. It applies a correction based on three terms:

• Proportional (P): Responds to the current error.
• Integral (I): Accumulates past errors to eliminate steady-state offset.
• Derivative (D): Predicts future error based on its rate of change.

While simpler and computationally lighter than MPC, PID is a reactive controller that lacks MPC's predictive, optimization-based horizon and explicit constraint handling. It is often used for simpler, single-input-single-output (SISO) loops where MPC's complexity is unnecessary.

Linear Quadratic Regulator (LQR) is an optimal state-feedback control strategy for linear systems. It minimizes a quadratic cost function that typically penalizes both state deviation from a reference and control effort over an infinite time horizon. The solution is a constant gain matrix, making it computationally efficient for runtime.

Key distinction from MPC: LQR provides an optimal, fixed control law for an unconstrained, linear system over an infinite horizon. MPC, in contrast, solves a finite-horizon, often constrained optimization problem online at each time step, allowing it to handle nonlinearities and explicit constraints (like actuator limits) directly.

Receding Horizon Control (RHC) is the fundamental operational principle of MPC. It describes the iterative process where, at each control interval:

1. The controller uses a dynamic model to predict the system's future behavior over a finite prediction horizon.
2. It solves an optimization problem to compute a sequence of optimal control inputs over that horizon.
3. Only the first control input from the optimized sequence is applied to the system.
4. The horizon "recedes" forward by one step, and the process repeats with new state measurements.

This feedback mechanism allows the controller to continuously adjust for model inaccuracies and disturbances.

Trajectory optimization is the broader mathematical field of computing a sequence of states and control inputs that minimizes a cost function while satisfying system dynamics and constraints. It is the core computational problem solved at each step of an MPC loop.

Common algorithms used within MPC solvers include:

• Sequential Quadratic Programming (SQP): Iteratively approximates the nonlinear problem as a quadratic program.
• Interior Point Methods: Handle constraints by incorporating them into a modified objective function.
• Differential Dynamic Programming (DDP): A dynamic programming approach efficient for unconstrained or lightly constrained problems. MPC can be viewed as online, closed-loop trajectory optimization.

State estimation is the process of inferring the internal, often unmeasurable, state variables of a system (e.g., velocity, friction coefficients) from noisy sensor data and a system model. Accurate state information is the critical input to the MPC's prediction step.

Key estimators used with MPC:

• Kalman Filter (KF): Optimal for linear systems with Gaussian noise.
• Extended Kalman Filter (EKF): Linearizes nonlinear models for state estimation.
• Unscented Kalman Filter (UKF): Uses a deterministic sampling approach for better nonlinear performance.
• Moving Horizon Estimation (MHE): The "dual" of MPC, solving a past-horizon optimization problem to estimate the state sequence that best fits recent measurements, offering explicit constraint handling.

A hard real-time constraint is a system requirement where missing a specified deadline is considered a total system failure. For MPC, this translates to a strict upper bound on the time allowed to solve the optimization problem within each control cycle.

Engineering implications for MPC:

• The optimization solver must guarantee completion within the allocated time slot (the Worst-Case Execution Time or WCET).
• This often necessitates using simplified models, fixed iteration count solvers, or explicit MPC (where the optimization is solved offline for all regions of state space, resulting in a simple lookup table online).
• The entire software pipeline, from sensor read to actuator command, must be analyzed for deterministic execution to ensure the MPC update is always delivered on time.