Model Predictive Control (MPC)

This is the defining temporal mechanism of MPC. At each control step, the controller solves an optimal control problem over a finite future window, or prediction horizon. Only the first control input from the optimized sequence is applied to the system. At the next time step, the horizon "recedes" forward, the optimization is re-solved with new state measurements, and the process repeats. This provides continuous feedback and adaptation.

• Core Mechanism: Solve, apply first step, re-measure, re-solve.
• Benefit: Inherently compensates for disturbances and model inaccuracies by frequently re-planning based on actual system state.

MPC's predictive capability is entirely dependent on an internal mathematical model of the system's dynamics. This model, which can be linear, nonlinear, or even data-driven (e.g., a neural network), is used to simulate and predict future system states in response to candidate control sequences.

• Model Types: Linear time-invariant (LTI) models are common for computational speed; Nonlinear models capture complex dynamics like aerodynamics or chemical reactions.
• Critical Role: The accuracy and fidelity of this model directly determine the controller's performance and robustness. Model mismatch is a primary source of error.

A major advantage of MPC is its ability to explicitly and systematically handle hard and soft constraints during the optimization. Constraints are not treated as afterthoughts but are baked directly into the solved problem.

• Input Constraints: Limits on actuator range (e.g., steering angle, throttle position).
• State Constraints: Safety or operational limits (e.g., maximum roll angle for a drone, minimum temperature in a reactor).
• Output Constraints: Limits on measured outputs. The optimizer finds the best control action that satisfies all these constraints over the prediction horizon, ensuring safe and feasible operation.

MPC naturally handles Multiple-Input, Multiple-Output (MIMO) systems. The optimization framework can simultaneously coordinate numerous actuators to control several interdependent output variables, managing the inherent couplings between them.

• Example: In an autonomous car, a single MPC controller can coordinate steering, throttle, and braking to simultaneously track a path (lateral control) and maintain a speed (longitudinal control), accounting for their interaction.
• Contrast: Traditional PID control often requires decoupled, single-loop designs that struggle with cross-coupling effects.

Control actions are not determined by a fixed control law (like u = -Kx in LQR) but are the solution to a numerical optimization problem. This problem minimizes a cost function (or objective function) over the prediction horizon.

• Typical Cost Function: J = Σ (tracking error)² + Σ (control effort)² + Σ (change in control)².
• Flexibility: The cost function can be tailored for specific performance goals: minimize energy, maximize comfort, prioritize certain errors, etc. This moves control design from tuning gains to engineering objectives.

MPC seamlessly combines feedforward and feedback control. The use of the model for prediction provides a feedforward element: it anticipates the system's response to future inputs. The receding horizon mechanism, which incorporates the latest measured state at each step, provides robust feedback correction.

• Feedforward: Based on the model's prediction of future states along the horizon.
• Feedback: The re-optimization at each step corrects for prediction errors caused by disturbances, noise, and model inaccuracy. This dual nature makes MPC highly effective for reference tracking and disturbance rejection.

The Linear Quadratic Regulator (LQR) is an optimal feedback controller for linear systems that minimizes an infinite-horizon quadratic cost function. It provides a closed-form, static gain matrix. MPC vs. LQR:

• LQR solves an infinite-horizon problem offline, resulting in a fixed control law.
• MPC solves a finite-horizon, often constrained problem online at each time step.
• Connection: LQR can be seen as the unconstrained, infinite-horizon special case of MPC. For linear systems with quadratic costs and no constraints, the MPC solution converges to the LQR solution as the prediction horizon extends to infinity.

Nonlinear Programming (NLP) is the mathematical process of optimizing an objective function subject to constraints, where the function or constraints are nonlinear. It is the computational engine of MPC.

• MPC as NLP: At each control step, MPC formulates and solves an NLP. The objective is the cost over the prediction horizon (e.g., tracking error, control effort). The constraints include the system's dynamic model, actuator limits, and safety boundaries.
• Solvers: Real-time MPC relies on efficient NLP solvers like IPOPT (Interior Point OPTimizer) or SNOPT (Sparse Nonlinear OPTimizer) to find solutions within the strict sampling time of the control loop (often milliseconds).

Sequential Quadratic Programming (SQP) is a leading iterative method for solving Nonlinear Programming (NLP) problems, making it a core algorithm for many MPC implementations.

• How it works: SQP approximates the original NLP at each iteration by a Quadratic Programming (QP) subproblem. This QP has a quadratic approximation of the Lagrangian (the cost) and linearized constraints.
• Role in MPC: For real-time MPC, real-time iteration (RTI) SQP is commonly used. It performs a single SQP iteration per control cycle, using the solution from the previous step as a warm start. This dramatically reduces computation time, making nonlinear MPC feasible for fast systems like autonomous vehicles or quadrotors.

Receding Horizon Control (RHC) is the operational principle that defines MPC's online execution loop. It is often used synonymously with MPC.

• The Algorithm:
  1. Measure/Estimate the current system state.
  2. Solve an optimal control problem over a finite future horizon, using the system model.
  3. Apply only the first control input from the computed optimal sequence.
  4. Shift the horizon forward by one time step and repeat.
• Key Benefit: This feedback mechanism provides inherent robustness to model inaccuracies and disturbances. By constantly re-planning based on the latest state, the controller can correct for prediction errors that accumulate over the horizon.

Trajectory Optimization is the broader mathematical framework for finding a sequence of states and control inputs that minimizes a cost function while satisfying dynamic constraints. MPC is a specific, online instance of this framework.

• Offline vs. Online: Classical trajectory optimization (e.g., for robot arm motion) is often solved once, offline, to generate a full trajectory. MPC performs online trajectory optimization repeatedly in a receding horizon fashion.
• Applications: While offline optimization plans a full path, MPC is used for feedback control, reacting to disturbances in real-time. They are complementary: an offline trajectory can provide the reference trajectory that an MPC controller tracks while handling real-world perturbations.

Controllability and Observability are fundamental system properties that critically impact the design and stability of any model-based controller, including MPC.

• Controllability: Determines whether the system's state can be driven to any desired value using admissible control inputs. An MPC controller cannot stabilize or control an uncontrollable system.
• Observability: Determines whether the system's internal state can be inferred from its outputs (measurements). Since MPC requires a state estimate (e.g., from a Kalman Filter), the system must be observable, or at least detectable, for the controller to function correctly. These properties are checked using the system's A and B (controllability) and A and C (observability) matrices in the linear case.