Receding Horizon Control

The defining characteristic is the rolling or shifting horizon. At each control time step k:

• An Optimal Control Problem (OCP) is solved over a finite prediction horizon (e.g., from time k to k+N).
• This yields an optimal sequence of future control inputs.
• Only the first control input of this sequence is applied to the physical system.
• At the next time step k+1, the horizon 'recedes' by one step. The OCP is solved again using the latest state measurement as the new initial condition. This creates a feedback mechanism that continuously corrects for model inaccuracies and unmeasured disturbances.

Unlike open-loop optimal control, RHC is inherently a closed-loop strategy. The periodic re-optimization based on new measurements provides disturbance rejection and robustness to model mismatch.

Key mechanism:

• The initial state for each optimization is updated with the latest state estimate (e.g., from a Kalman Filter).
• This feedback corrects the predicted trajectory, preventing the controller from blindly following a plan made with outdated or inaccurate information.
• It transforms a finite-horizon open-loop optimization into an infinite-horizon closed-loop control law in practice.

RHC requires solving a numerical optimization problem in real-time, within one sampling period of the control system. This is a hard real-time computational constraint.

Solver types depend on the problem formulation:

• Linear MPC / Quadratic Programming (QP): Uses a linear model and quadratic cost. Solved with active-set or interior-point QP solvers.
• Nonlinear MPC (NMPC): Uses nonlinear models/costs. Solved with Sequential Quadratic Programming (SQP) or Interior-Point Methods.
• Explicit MPC is an alternative that pre-computes the solution offline as a piecewise affine function, trading online computation for memory.

A primary advantage of RHC is the explicit, forward-looking handling of constraints on states and control inputs.

Constraint types:

• Hard Constraints: Physical limits that must never be violated (e.g., actuator saturation, safe temperature ranges). Enforced directly in the optimization.
• Soft Constraints: Operational limits that can be temporarily breached at a cost (e.g., slight overshoot of a setpoint). Implemented using slack variables in the cost function to ensure optimization feasibility.
• Terminal Constraints: Often used in stability proofs, requiring the final predicted state to lie in a terminal set.

The internal dynamic model is the core of the prediction. Its accuracy directly dictates controller performance.

Common model forms:

• Linear Time-Invariant (LTI): State-space models (e.g., x(k+1) = Ax(k) + Bu(k)). Used in Linear MPC.
• Nonlinear Ordinary Differential Equations (ODEs): dx/dt = f(x, u). Used in NMPC for robotics, chemical processes.
• Discrete-time models: Derived from continuous models via sampling. The model is used to simulate/predict future system states over the horizon, given a sequence of candidate control inputs. System Identification techniques are often used to build this model from data.

The choice of prediction horizon length (N) is a critical design parameter with a stability trade-off.

Short Horizon (N small):

• Lower computational cost.
• Can lead to myopic behavior and instability, as the controller doesn't 'see' far enough to make stable long-term decisions.

Long Horizon (N large):

• Better performance and typically easier to guarantee stability.
• Significantly higher computational burden.

Stability Enforcement: To guarantee closed-loop stability with a finite horizon, designers often use:

• Terminal Cost: A cost term penalizing the final state (often based on the Algebraic Riccati Equation solution).
• Terminal Constraint: Forcing the final predicted state into a pre-defined stabilizing set.
• Constraint Tightening: In Robust MPC, to account for uncertainties.

RHC is the standard framework for the local planning and control layer of self-driving cars. At each time step (e.g., every 100ms), the controller:

• Predicts the future states of the ego vehicle and surrounding traffic over a 3-10 second horizon.
• Solves an optimization for a smooth, collision-free trajectory that respects vehicle dynamics (e.g., bicycle model), actuator limits (steering rate, acceleration), and traffic rules.
• Applies the first optimal steering and acceleration commands before re-planning with new sensor data. This enables safe lane-keeping, adaptive cruise control, and complex maneuvers like merging in dense, unpredictable traffic.

In continuous industrial processes like chemical reactors or distillation columns, Economic MPC using RHC maximizes profit or minimizes energy use while enforcing critical safety constraints.

• The internal model predicts product composition, temperatures, and pressures.
• The cost function directly encodes economic objectives (e.g., maximize yield of a high-value product, minimize steam consumption).
• Hard constraints on temperature, pressure, and valve positions ensure safe operation within the plant's operating envelope.
• The receding horizon allows the controller to continuously adjust to feedstock variations, catalyst degradation, and changing market prices, optimizing the process in real-time.

RHC provides the aggressive, stable flight control required for unmanned aerial vehicles (UAVs) and spacecraft. Nonlinear MPC (NMPC) handles the complex, underactuated dynamics.

• For a quadrotor, the model includes nonlinear rigid-body dynamics and aerodynamics.
• The controller solves for optimal motor thrusts to track a desired position and attitude trajectory while avoiding obstacles.
• Constraint handling is vital for respecting motor saturation limits and keeping the vehicle within a defined geofence.
• The receding implementation provides robustness to wind gusts and modeling errors, enabling precise maneuvers like flying through narrow windows or perching.

Robotic arms use RHC for dynamic manipulation tasks where contact forces and object interactions must be predicted and controlled.

• The internal model combines the arm's dynamics with a model of the manipulated object (e.g., its inertia, friction).
• The optimization computes joint torques to achieve a task like inserting a peg into a hole, pushing an object, or catching a ball, while respecting torque limits and self-collision constraints.
• The receding horizon allows the controller to react in real-time to slippage, unexpected contact forces, or perturbations, making the manipulation robust and adaptive.

RHC optimizes energy flows in complex systems with storage, generation, and time-varying demand/price. Key applications include:

• Building Climate Control: Minimizes HVAC energy cost over a day while maintaining comfort zones, predicting weather and occupancy.
• Microgrid Management: Schedules dispatch from batteries, solar panels, and generators to meet demand, using forecasts for renewable generation and electricity prices.
• Electric Vehicle Charging: Optimizes charging schedules for a fleet to minimize cost and grid impact, subject to each vehicle's departure time and battery constraints. The receding horizon continuously incorporates updated forecasts and real-time measurements, adapting to prediction errors.

For bipedal and quadruped robots walking on rough terrain, RHC (specifically Nonlinear MPC) is used for whole-body dynamics control.

• A simplified model (e.g., Single Rigid Body or Linear Inverted Pendulum) predicts the robot's center-of-mass dynamics.
• At each control cycle, the optimization solves for optimal ground reaction forces and footstep placements for the next several steps.
• It enforces kinematic constraints (leg length, joint limits) and dynamics constraints (friction cones, torque limits).
• Applying only the immediate force commands and then re-planning allows the robot to recover from pushes, step on moving platforms, and traverse uneven ground dynamically and stably.

The Optimal Control Problem (OCP) is the core mathematical formulation solved at each step of Receding Horizon Control. It is defined by three key components:

• A dynamic model (e.g., differential equations) that predicts future system states.
• A cost function to be minimized, penalizing tracking error and control effort.
• A set of constraints on system states and control inputs (e.g., actuator limits, safety boundaries). The OCP is solved repeatedly over a moving time window to generate the optimal control sequence.

These are the two critical time parameters defining the Receding Horizon Control window.

• Prediction Horizon (N): The future time window over which the system's behavior is predicted using the internal model. A longer horizon improves performance but increases computational cost.
• Control Horizon (M): The (often shorter) window over which control moves are optimized. Beyond M, control inputs are typically held constant or set to zero. The distinction allows for a trade-off between performance and computational tractability.

Moving Horizon Estimation (MHE) is the dual problem to Receding Horizon Control. While MPC optimizes future control inputs, MHE optimizes past state estimates. It solves a constrained optimization problem over a sliding window of recent measurements to provide the most likely current system state, which serves as the crucial initial condition for the MPC's prediction. MHE explicitly handles sensor noise, model uncertainty, and state constraints.

Real-Time Optimization (RTO) refers to the stringent computational requirement of Receding Horizon Control. The OCP must be solved within one sampling period of the physical system (often milliseconds) to compute the next control action. This demands highly efficient numerical solvers (e.g., Quadratic Programming (QP) solvers for linear MPC, Sequential Quadratic Programming (SQP) for nonlinear MPC) and techniques like warm-starting to accelerate convergence.

These are advanced design elements used to guarantee the long-term stability of the Receding Horizon Control loop.

• Terminal Cost: An additional term in the OCP's cost function evaluated at the final state of the prediction horizon. It often approximates the cost-to-go from that state to infinity.
• Terminal Constraint: A requirement that the predicted state at the end of the horizon must lie within a pre-defined stabilizing set. Together, they ensure the finite-horizon optimization approximates an infinite-horizon solution, preventing short-sighted control actions.

A warm start is a critical performance optimization for the online solver in Receding Horizon Control. Instead of initializing the optimization from scratch at each time step, the solution from the previous step (shifted in time) is used as the initial guess. This dramatically reduces the number of iterations required for convergence, especially for Nonlinear MPC (NMPC), making it feasible to meet real-time optimization deadlines on embedded hardware.