Committed Horizon Control Approach

• Committed horizon control is a dynamic model predictive strategy that adaptively adjusts the optimization horizon and commits to a control sequence for improved stability and performance.
• It integrates horizon adaptation rules, suboptimality guarantees, and constraint synchronization to effectively handle nonlinear, linear, and hybrid systems.
• Practical implementations in PDE control, DDP, and MDP frameworks demonstrate reduced computational load and enhanced control performance compared to fixed-horizon methods.

A committed horizon control approach is a class of model predictive and receding horizon control strategies characterized by adaptively selecting, at each step or decision point, a finite optimization horizon (the "committed horizon") and then strictly executing a control sequence or policy over a prescribed portion (or entirety) of that horizon before the next re-optimization is performed. This paradigm provides theoretical and computational advantages in terms of suboptimality guarantees, stability, feasibility under constraints, and adaptability to varying system requirements, extending classical fixed-horizon MPC to variable- or on-line adaptive-horizon frameworks. Committed horizon control is systematically developed for nonlinear, linear, and hybrid control systems, and encompasses applications ranging from optimal PDE control to constrained nonlinear dynamical systems and Markov decision processes (Pannek, 2011, Braun et al., 2014, Arvelo et al., 2012, Chang, 2022, Stachowicz et al., 2021, Xiao et al., 2018).

1. Foundational Principles of Committed Horizon Control

Committed horizon control generalizes the canonical predictive control scheme by dynamically adjusting the optimization (prediction) or control (holding/execution) horizon as part of the closed-loop feedback algorithm. At every decision epoch (e.g., a sampling step), the controller may either:

• Adapt the prediction/optimization horizon: Shortening when the current choice is more than sufficient for performance/stability, or prolonging if needed to maintain a prescribed bound on suboptimality, feasibility, or constraint satisfaction (Pannek, 2011, Braun et al., 2014).
• Commit control over a selected interval: Executing the computed open-loop control sequence during a "committed period" or control horizon, during which re-planning may not be performed except under specified conditions (e.g., robustness triggers) (Braun et al., 2014, Arvelo et al., 2012).

Key objectives and guarantees include:

• Asymptotic stability or recursive feasibility, even in the absence of terminal costs or constraints (Braun et al., 2014).
• Uniform suboptimality bounds—achieving a given ratio between closed-loop and infinite-horizon costs (Pannek, 2011).
• Explicit handling of complex constraints (e.g., interval-wise energy budgets) by synchronizing the optimization window with constraint intervals (Arvelo et al., 2012).

2. Mathematical Formulation and Algorithmic Structure

The essential structure of committed horizon control can be formalized for discrete-time nonlinear systems: $x_{k+1} = f(x_k, u_k),\quad x_k\in X,\;u_k\in U$ Let $N_k$ denote the optimization horizon chosen at step $k$, and consider the following finite-horizon cost: $$J_{N_k}(x_k) = \min_{u_{0:N_k-1}} \sum_{i=0}^{N_k-1} \ell(x_{k+i}, u_{k+i}) + V_f(x_{k+N_k})$$ Here $\ell$ is the stage cost, $V_f$ is a terminal cost, and constraints or additional structure (e.g., terminal region, interval wise constraints) are enforced as needed (Pannek, 2011, Arvelo et al., 2012).

An outline of the committed horizon MPC algorithm is:

1. Measure the current state $x_k$.
2. Choose a candidate horizon $N_k$ (possibly inherited or adapted from $N_{k-1}$).
3. Solve the OCP with horizon $N_k$ to obtain optimal inputs and suboptimality measure $\alpha(N_k)$.
4. If $\alpha(N_k)\geq \bar\alpha$ (desired suboptimality), attempt to shorten $N_k$. If not, prolong $N_k$ via a suboptimality-increasing map or fixed-point rule.
5. Commit: apply the first input (or commit to a subinterval according to the control horizon).
6. Advance $k$ and repeat (Pannek, 2011, Braun et al., 2014).

For PDE control or DDP-based approaches, similar solve–apply–shift or backward/forward sweep cycles are executed over adaptive windows (Xiao et al., 2018, Stachowicz et al., 2021).

3. Horizon Adaptation, Commitment, and Suboptimality Guarantees

The principal innovation is adaptive/on-line horizon selection and explicit commitment to a computed control sequence for a window that is only as long as required to maintain performance and feasibility:

• Shortening rules (a posteriori or a priori): If Lyapunov or suboptimality inequalities are satisfied over the planned trajectory, the horizon can be reduced while maintaining guarantees. Example: if

$$V_{N_k}(x_k) \geq V_{N_k}(x_{k+1}) + \bar\alpha\,\ell(x_k,u_k)$$

holds for $N_k-1$, then commit to shortening (Pannek, 2011, Braun et al., 2014).

• Prolongation rules: If the suboptimality bound is violated, increment $N_k$ stepwise or using monotonic maps/fixed-point iterations until the bound is restored (Pannek, 2011).
• Performance bounds: Under a growth/decay assumption on the value function, the closed-loop cost satisfies

$$V_\infty^{\mathrm{MPC}}(x) \leq \alpha_{T,\delta}^{-1} V_\infty(x)$$

for prediction horizon $T$ and control (commitment) horizon $\delta$ (Braun et al., 2014).

For interval-wise constrained systems, the optimization horizon is synchronized with the constraint intervals, and the controller commits to an input sequence that satisfies the cumulative constraint over the entire interval (Arvelo et al., 2012).

4. Practical Algorithmic Implementations

Different formulations of the committed horizon concept are implemented in the literature:

• Nonlinear PDE control: Successive, windowed finite-horizon OCPs are solved using adjoint-based gradient methods. At each window, state variables are checkpointed, forward/backward sweeps are performed, and the control law is updated by gradient descent, then applied non-overlappingly over the committed interval (Xiao et al., 2018).
• Differential Dynamic Programming (DDP): The horizon length $T$ is treated as a decision variable within DDP recursions. Quadratic expansions of the value function are evaluated for candidate horizons in a window around the current one, and the one minimizing the local cost-to-go is “committed” at each iteration. The local gains define the closed-loop law over the interval (Stachowicz et al., 2021).
• Markov Decision Processes: In finite MDPs, the forecast-horizon policy is optimized online at the visited state using asynchronous policy improvement and policy-switching over a rolling horizon. Supervisor feedback or policy proposals can be integrated. Convergence to an optimal committed policy is guaranteed under mild communication assumptions (Chang, 2022).
• Interval-wise constrained systems: The horizon variable jumps forward in blocks to align with energy or other interval-wise constraints, enforcing contractive terminal constraints and committing to N-step input sequences over entire constraint windows (Arvelo et al., 2012).

5. Stability, Feasibility, and Robustness

Committed horizon methods provide strong guarantees:

• Stability: Sufficient decrease of the value function along the closed loop, enforced via Lyapunov-type inequalities, ensures asymptotic convergence to the desired setpoint or invariant set, even with variable horizons (Pannek, 2011, Braun et al., 2014).
• Recursive Feasibility: By virtue of online adaptation and by only shortening if preceding solutions remain feasible for reduced horizons, recursive feasibility is maintained (Pannek, 2011).
• Robustness Enhancements: Slack-based schemes can tolerate local, temporary violations of performance constraints, provided that cumulative performance over time is preserved. Intermediate updates inside the commit window can be triggered by Lyapunov-type update conditions, improving robustness to model mismatch or disturbances (Braun et al., 2014).
• Constraint Satisfaction: For structured constraints (e.g., interval-wise energy budgets), the optimization horizon is structured to exactly cover the constrained interval, and the committed policy ensures no exceedance by construction (Arvelo et al., 2012).

6. Applications and Comparative Results

Committed horizon control strategies have demonstrated effectiveness in several domains:

• Flow control: Nonlinear optimal control of bypass transition in boundary layers using receding/committed horizon windows leads to significant drag reduction, with time-averaged statistics and spatially varying actuation laws revealed by the approach (Xiao et al., 2018).
• Energy-constrained systems: In systems with interval-wise total energy constraints, interval-synchronized committed horizon MPC with contractive final-state constraints achieves stability and respects energy limits, outperforming traditional fixed-horizon receding horizon controllers both in stability and cost in simulations (Arvelo et al., 2012).
• Nonlinear and high-dimensional MPC: Adaptive-horizon MPC variants use substantially smaller horizons, maintaining tracking performance while reducing computational cost compared to long fixed-horizon MPC. The a posteriori horizon adaptation usually yields the smallest average horizon and shortest CPU time (Pannek, 2011, Braun et al., 2014).
• Finite-horizon MDPs: Rolling-horizon policy iteration with policy-switching achieves monotonic value improvement at visited states, with finite-time global optimality under communication, and allows online incorporation of supervisor-provided candidate policies (Chang, 2022).

7. Challenges, Limitations, and Design Considerations

Key challenges include:

• Computational Load: Each committed horizon interval may require repeated forward/adjoint solves (in PDEs) or multiple QP/NLP re-solves (in nonlinear MPC), imposing high storage and computational burdens, especially for large systems or long horizons (Xiao et al., 2018, Pannek, 2011).
• Horizon Selection Tradeoffs: The choice of prediction and commit horizons balances stability, computational feasibility, and control performance. Excessively short horizons can destroy stability; excessively long horizons incur unnecessary computation (Braun et al., 2014, Stachowicz et al., 2021).
• Parameter Tuning: Step lengths, suboptimality bounds, and convergence tolerances must be set judiciously to guarantee monotonic cost decrease and avoid oscillatory or unstable horizon adaptation (Xiao et al., 2018, Pannek, 2011).
• Constraint Binding: For complex or non-periodic constraints, construction of contractive terminal sets or suitable synchronization of the commit window may become nontrivial and may limit applicability (Arvelo et al., 2012).
• Global vs. Local Optimality: Committed horizon strategies provide local (per-interval) optimality; global closed-loop optimality is traded for computational tractability and robustness to chaotic sensitivities (particularly in nonlinear and PDE systems) (Xiao et al., 2018, Stachowicz et al., 2021).

A plausible implication is that as computational and algorithmic advances continue, the utility of committed horizon and adaptive-horizon MPC approaches will further expand in practice, particularly for systems where fixed-horizon MPC is infeasible due to complexity, constraint structure, or real-time requirements.