Prediction Horizon and Control Horizon

The prediction horizon is the future time window over which the MPC controller predicts the system's behavior using its internal dynamic model. It defines how far into the future the controller 'looks ahead' to anticipate the consequences of its actions.

• Purpose: Enables proactive, anticipatory control by accounting for future reference trajectories, disturbances, and constraints.
• Trade-off: A longer horizon generally improves stability and performance but increases computational complexity. A horizon that is too short can lead to myopic, unstable control.
• Typical Values: Ranges from tens of time steps for fast mechanical systems (e.g., drone attitude control) to hundreds for slow chemical processes.

The control horizon is the (typically shorter) time window over which the MPC optimizer computes a sequence of optimal control moves. Beyond this horizon, control inputs are usually assumed constant or zero.

• Purpose: Reduces the dimensionality of the optimization problem. The optimizer solves for Nc control moves instead of Np, drastically cutting computation.
• Effect: After the control horizon, the system evolves in open-loop based on the last computed input. This is a key simplification.
• Rule of Thumb: Nc is often chosen as 10-20% of Np, or long enough to capture the system's dominant dynamics.

This is the core MPC execution loop. At each sampling instant:

1. Measure/estimate the current system state.
2. Solve an Optimal Control Problem (OCP) over the prediction horizon, optimizing control inputs over the control horizon.
3. Apply only the first control input from the optimized sequence to the actual system.
4. Shift the horizons forward by one time step and repeat with new measurements.

This feedback mechanism provides robustness to model inaccuracies and disturbances.

Selecting Np and Nc is a critical design choice:

• Prediction Horizon (Np): Must be long enough to capture the settling time of the system. A good starting point is Np * Ts (sample time) ≈ 5-10 times the open-loop settling time.
• Control Horizon (Nc): Must be long enough to give the optimizer sufficient 'authority' to steer the predicted trajectory. If Nc is too short, performance degrades.
• Computational Cost: The online optimization problem size scales with Nc. For Nonlinear MPC (NMPC), even a small increase in Nc can make real-time execution infeasible.

Horizon lengths directly affect closed-loop guarantees:

• Stability: For Linear MPC, a sufficiently long prediction horizon with a terminal cost (often from LQR) and/or a terminal constraint can guarantee nominal stability.
• Performance: A longer Np allows better tracking of future reference changes and rejection of slow disturbances. A well-chosen Nc ensures the optimizer can achieve this performance.
• Constraint Handling: The prediction horizon Np must be long enough to 'see' future constraint violations (e.g., an approaching obstacle or tank limit) to avoid them proactively.

Consider an MPC controller for lane-keeping:

• Sample Time (Ts): 0.1 seconds.
• Prediction Horizon (Np): 30 steps → Looks ahead 3 seconds. This is enough to see the road curvature.
• Control Horizon (Nc): 10 steps → Optimizes steering angle for the next 1 second, then assumes it holds steady.
• Process: Every 0.1s, the MPC solves for the best steering over the next 1s to stay centered over the next 3s, applies the first command, and repeats. This balances smooth control with computational limits.

This is the fundamental operating principle of MPC. At each control step, the controller:

• Solves an Optimal Control Problem (OCP) over the prediction horizon.
• Applies only the first control input from the optimized sequence to the physical system.
• Shifts the horizon forward by one time step, incorporates new sensor feedback, and repeats the optimization. This rolling implementation makes MPC a feedback control law, allowing it to correct for model inaccuracies and disturbances.

The OCP is the mathematical core solved at each MPC step. Its formulation is directly defined by the horizons:

• Prediction Horizon (Np): Sets the time window [t, t+Np] for the problem's dynamics and constraints.
• Control Horizon (Nc): Defines the window [t, t+Nc] over which control variables are decision variables. The OCP is defined by a dynamic model (for prediction), a cost function (to minimize), and state/input constraints, all evaluated over these horizons.

These are advanced design tools used to ensure closed-loop stability, especially with a finite prediction horizon.

• Terminal Cost: An additional term V(x(t+Np)) added to the cost function, penalizing the state at the end of the prediction horizon. It often approximates the cost-to-go from that point onward.
• Terminal Constraint: A requirement that the predicted state at t+Np must lie within a terminal set X_f. This set is designed to be positively invariant. Together, they act as a surrogate for an infinite horizon, guaranteeing stability without needing to predict infinitely far into the future.

This is the critical runtime requirement for MPC. The OCP defined by the horizons must be solved within the controller's sampling period (e.g., milliseconds).

• The length of the horizons (Np, Nc) directly impacts problem size and solver time.
• Linear MPC with a Quadratic Program (QP) can use fast, dedicated QP solvers.
• Nonlinear MPC (NMPC) requires more complex Nonlinear Programming (NLP) solvers, like Sequential Quadratic Programming (SQP), making horizon selection a key trade-off between performance and computational feasibility.

MHE is the dual problem to MPC, used for state estimation. It also employs a horizon-based optimization.

• Estimation Horizon: A window of past measurements and inputs used to solve for the most likely current state.
• Like MPC, it uses a system model and can explicitly handle constraints and nonlinearities.
• The estimation horizon length is a tuning parameter balancing noise filtering, computational load, and memory of past data. MHE provides the initial state x(t) for the MPC's prediction horizon.

A critical technique for meeting RTO demands, especially in Nonlinear MPC. It leverages the temporal structure imposed by the receding horizon.

• The solution from the previous time step (over horizon [t-1, t-1+Np]) is shifted and used as the initial guess for the optimizer at the current step (horizon [t, t+Np]).
• Since the predicted trajectories between consecutive steps are often similar, this provides a high-quality initial guess.
• This dramatically reduces the number of solver iterations needed, enabling the use of longer prediction and control horizons.