Model Predictive Control (MPC)

This is the core execution loop of MPC. The controller solves an optimization problem over a finite prediction horizon (e.g., the next 10 seconds) to determine an optimal sequence of future control inputs. However, only the first control action from this sequence is applied to the physical system. At the next time step, the horizon 'recedes' forward, new measurements are taken, and the optimization is repeated with updated information. This closed-loop feedback mechanism allows MPC to continuously correct for model inaccuracies and unmeasured disturbances.

A major advantage of MPC is its ability to directly incorporate hard and soft constraints into the optimization problem. This is a fundamental improvement over traditional PID controllers, which handle constraints indirectly (e.g., through anti-windup). Common constraints include:

• Actuator Limits: Maximum/minimum force, torque, voltage, or valve position.
• State Constraints: Safe operating ranges for temperature, pressure, velocity, or position.
• Output Constraints: Limits on process variables like product purity or concentration. The optimizer finds the best control sequence that satisfies these constraints, ensuring safe and feasible operation.

MPC relies on an internal model—mathematical or learned—to predict the future states of the system over the horizon. Model fidelity is critical for performance.

• First-Principles Models: Often derived from physics (e.g., differential equations for chemical kinetics or vehicle dynamics).
• Data-Driven Models: Linear time-invariant (LTI) models identified from process data, or nonlinear models like Neural Networks or Gaussian Processes.
• Hybrid Models: Combine physical knowledge with data-driven components. The accuracy of this model directly determines the controller's ability to anticipate system behavior and optimize accordingly.

At each control interval, MPC solves a (often constrained) optimization problem in real-time. The objective function typically penalizes:

• Tracking Error: Deviation from a desired reference trajectory.
• Control Effort: Magnitude or change in control actions to promote smooth operation.
• Terminal Cost: A penalty on the predicted state at the end of the horizon to ensure stability. Solving this problem requires significant computational power. For linear systems with quadratic costs, it becomes a Quadratic Program (QP). For nonlinear systems, it is a more complex Nonlinear Program (NLP), pushing the limits of real-time solvers.

MPC naturally handles Multiple-Input, Multiple-Output (MIMO) systems. The internal model captures the complex couplings between different actuators and process variables. The optimizer simultaneously calculates moves for all manipulated variables to achieve the best overall performance for all controlled outputs. This is a stark contrast to deploying multiple, decoupled single-loop PID controllers, which often fight each other due to unaccounted interactions. This makes MPC the standard for complex, interconnected processes like chemical distillation columns or autonomous vehicle trajectory tracking.

MPC can proactively compensate for measured disturbances and known future changes in the reference (feedforward control). The internal model predicts how these known disturbances will affect the future states, and the optimizer preemptively adjusts the control inputs to counteract them. Furthermore, by comparing the model's predicted output with the actual measured output from sensors, an implicit disturbance estimate is generated. This feedback correction is integral to the receding horizon loop, providing robust rejection of unmeasured disturbances and robustness to steady-state model error.

A reinforcement learning paradigm where an agent learns or is given an internal model of the environment's dynamics. This model is used for planning and policy improvement, often through simulated rollouts. Unlike MPC, which typically uses a known or identified model, MBRL focuses on learning the model from interaction data. Key approaches include:

• Dyna-style architectures that interleave real experience with model-based planning.
• Model-based value expansion for more accurate value estimation.
• Purely learned world models used for imagination-based training.

An optimal control solution for linear systems with quadratic cost functions. LQR provides a closed-form, feedback control law (u = -Kx) that is optimal over an infinite horizon. It is a foundational benchmark and a special case within MPC:

• MPC vs. LQR: MPC handles constraints (e.g., torque limits, safe zones) explicitly, while LQR does not. MPC operates over a finite horizon and re-solves online, whereas LQR computes a single, constant gain matrix offline.
• iLQR (Iterative LQR): An extension that linearizes nonlinear dynamics around a trajectory and iteratively solves the LQR problem, forming the basis for many nonlinear MPC solvers.

The core operational principle of MPC. At each control step, the algorithm:

1. Measures the current state of the system.
2. Solves an optimization problem to find the best sequence of future control inputs over a finite prediction horizon.
3. Executes only the first control input from the optimized sequence.
4. Recedes the horizon forward by one step and repeats. This feedback mechanism provides robustness to model inaccuracies and external disturbances. The length of the prediction horizon is a critical trade-off between computational cost and controller performance.

The broader mathematical field concerned with finding a control law for a dynamical system that minimizes a cost function (or maximizes a performance index). MPC is a subset of optimal control methods characterized by its online, constrained, finite-horizon approach. Other branches include:

• Pontryagin's Maximum Principle: Provides necessary conditions for optimality in continuous-time problems.
• Dynamic Programming: Solves the problem by breaking it into simpler sub-problems (Bellman's principle), often facing the curse of dimensionality.
• Direct & Indirect Methods: Numerical techniques for solving optimal control problems, which MPC solvers often leverage.

A model-based control strategy, common in robotics, that computes the joint torques required to achieve a desired acceleration. It effectively linearizes and decouples the nonlinear robot dynamics. Relationship to MPC:

• MPC often operates at a higher level (e.g., planning Cartesian end-effector trajectories), while an inverse dynamics controller executes at a lower level to track those trajectories with high bandwidth.
• Combined Approach: MPC can generate optimal reference accelerations, which are then passed to an inverse dynamics block to compute the final joint-level commands, creating a hierarchical control architecture.

The computational engine at the heart of most practical MPC implementations. Since MPC requires solving a constrained optimization problem at every control cycle (often within milliseconds), the choice of solver is critical. Common classes include:

• Sequential Quadratic Programming (SQP): Iteratively approximates the NLP as a quadratic program.
• Interior-Point Methods: Handle constraints by incorporating barrier functions.
• Real-Time Iteration (RTI): A variant of SQP that performs only one SQP iteration per control step, trading optimality for speed.
• Code Generation Tools: Frameworks like ACADO or CasADi auto-generate optimized C code for the specific MPC problem, enabling deployment on embedded hardware.