Receding-Horizon Nonlinear MPC

• Receding-horizon nonlinear MPC is a control strategy that solves finite-horizon optimization problems to manage nonlinear dynamics and constraints in real-time.
• It employs terminal costs, constraint sets, and feedback policies to ensure closed-loop stability and safety, even under nonconvex conditions.
• Empirical results, such as in spacecraft attitude regulation, demonstrate reduced computation times and robust convergence with guaranteed state invariance.

Receding-horizon nonlinear model predictive control (MPC) is a class of feedback control strategies for nonlinear systems subject to constraints, in which control actions are computed by repeatedly solving optimal control problems over finite prediction horizons shifted forward as new state information arrives. This approach enables the explicit treatment of nonlinear dynamics, nonconvex state/input constraints, and stability or safety specifications through the cost and constraint structure of the optimization problem, while leveraging computational resources to approximate infinite-horizon or safety-critical control policies in a tractable, real-time fashion.

1. Formulation of the Receding-Horizon Nonlinear MPC Problem

Let the plant dynamics be given by either continuous- or discrete-time nonlinear equations:

• Continuous-time: $\dot x(t)=f(x(t),u(t))$ with $x(t)\in\mathcal C\subset\mathbb R^n$, $u(t)\in\mathcal U\subset\mathbb R^m$
• Discrete-time: $x_{k+1}=f(x_k, u_k)$ with $x_k\in\mathcal X$, $u_k\in\mathcal U$

State and input constraints are imposed over the horizon, typically of the form $x(t)\in\mathcal C$ and $u(t)\in\mathcal U$, or $g(x_t,u_t)\le0$.

The receding-horizon optimization is to minimize a cost functional over control actions $u(\cdot)$ over horizon $[0,T]$: $$J_{[0,T]}(x(\cdot),u(\cdot)) = F(x(T)) + \int_0^T L(x(t), u(t))\, dt$$ where $L$ is stage cost and $F$ is a terminal cost. The receding-horizon controller at time $t_k$ solves this optimal control problem (OCP) subject to current state $x_k:=x(t_k)$, applies the first part of $u^*$ for a short sampling period, then repeats the process as new measurements are obtained (Olucak et al., 26 Sep 2025).

A similar formulation holds in discrete time, with horizon length $N$ and cost

$$J_N(x_0,u_{0:N-1}) = F(x_N) + \sum_{k=0}^{N-1} L(x_k, u_k)$$

Constraints and terminal requirements can enforce stability or state invariance, as detailed below.

2. Stability, Invariance, and the Role of Terminal Ingredients

Nonlinear receding-horizon MPC schemes achieve closed-loop stability and constraint satisfaction by embedding terminal cost $F$, terminal constraint sets $\mathcal X_f \subset \mathcal C$, and—when possible—terminal state feedbacks $\kappa(x)$, such that for all $x\in \mathcal X_f$ there exists a control satisfying

$$\langle \nabla F(x), f(x, \kappa(x)) \rangle \le -L(x, \kappa(x)), \quad f(x, \kappa(x))\in T_x\mathcal X_f$$

guaranteeing invariance of $\mathcal X_f$ and dissipation/decay of $F$ (Olucak et al., 26 Sep 2025).

The optimal value function $V^*$ for the receding-horizon OCP acts as a control Lyapunov function (CLF), and the feasible set $\mathcal F$ can be characterized as a control barrier function (CBF) set. These satisfy:

• CLF: For all $x\in\mathcal F$, $$\inf_{u\in\mathcal U}\left\{\langle \nabla V^*(x), f(x,u)\rangle + L(x,u) \right\} \le 0$$
• CBF: For any $C^1$ function $h$ such that $\mathcal F = \{x: h(x)\le0\}$, one may enforce, for class-$\mathcal K$ function $\alpha$, $$\inf_{u\in\mathcal U} \langle \nabla h(x), f(x,u)\rangle \le \alpha(-h(x))$$ (Olucak et al., 26 Sep 2025).

These properties ensure that $V^*$ and $h$ verify stability and forward invariance in the region of attraction of the MPC.

3. Approximate and Infinitesimal-Horizon Receding-Horizon Control

Given the computational complexity of solving full-horizon nonlinear OCPs (often performed via large-scale nonlinear programming), recent research establishes that well-chosen approximations $\hat V$ and $\hat h$ (e.g., low-degree polynomials or sums-of-squares constructions) sufficing the same CLF/CBF dissipation and barrier conditions enable approximate receding-horizon nonlinear MPC with provable safety and convergence: $$\begin{aligned} &\langle \nabla\hat V(x), f(x,\hat\kappa(x))\rangle + L(x,\hat\kappa(x)) \le -\gamma(\|x\|) \ &\langle \nabla\hat h(x), f(x,\hat\kappa(x))\rangle \le \alpha(-\hat h(x)) \end{aligned}$$ for $\gamma,\alpha\in \mathcal K$, on an inner-approximate invariant set $\hat{\mathcal F}\subset\mathcal F$ (Olucak et al., 26 Sep 2025).

A key development is the replacement of the full-horizon MPC OCP by an infinitesimal-horizon quadratic program (termed “$\partial$MPC” QP) at each state: $$\begin{aligned} \kappa_{\partial}(x) = \arg\min_{u\in\mathcal U} L(x,u) + \langle \nabla\hat V(x), f(x, u) \rangle \ \text{subject to}\quad \langle \nabla\hat h(x), f(x,u)\rangle \le \alpha(-\hat h(x)) \end{aligned}$$ This QP is always feasible, guarantees asymptotic stability, and enforces forward invariance of the set $\hat{\mathcal F}$ provided the terminal-like inequalities are satisfied. SOS programming may be used to synthesize such approximations (Olucak et al., 26 Sep 2025).

4. Algorithmic Implementation and Real-Time Feasibility

The “$\partial$MPC” approach, operating via a single small QP per state evaluation, achieves several orders of magnitude lower computational cost than full-horizon NMPC. In spacecraft attitude regulation examples, solve times for the one-shot QP are $\sim$3 μs average, compared to $\sim$1\,s for full-horizon (Ipopt-based) NMPC and $50$ ms for RTI (Real-Time Iteration) sequential quadratic programming. With polynomial approximations for $\hat\kappa(x)$, evaluation is instantaneous ($\sim$0 μs) (Olucak et al., 26 Sep 2025).

Empirical results for a Hubble-like satellite subject to state and input limits, including nonconvex “keep out” pointing constraints, demonstrated:

• For a large-angle slew, the $\partial$MPC scheme ran at 10 Hz, kept the spacecraft safe, and achieved convergence with a lower total stage-cost integral than classical NMPC.
• Across 100 Monte Carlo trials with random initial states, all trajectories remained safe and converged under the QP controller, with worst-case solve time $<$ 0.1 ms.

These results highlight the strong real-time feasibility and robust safety properties of the “safe-by-design” receding-horizon NMPC methodology.

5. Theoretical Safety and Asymptotic Convergence

Provided that the approximate value and barrier functions satisfy the terminal-like dissipation and barrier inequalities, the closed-loop system controlled by the infinitesimal-horizon QP is:

• Forward invariant: If $x(0)\in\hat{\mathcal F}$, all $x(t)\in\hat{\mathcal F}$ for $t\geq0$,
• Asymptotically stable: The Lyapunov derivative $\dot{\hat V}(x(t)) \le -\gamma(\|x\|) < 0$ for $x\neq0$, implying $x(t)\to0$,
• Robust to approximation error: Approximate solutions still yield a valid CLF/CBF pair and hence safety/stability as long as the inequalities are certified (Olucak et al., 26 Sep 2025).

This framework generalizes to inner-approximate feasible sets and value functions obtained via system identification, learning, or polynomial/SOS optimizations, and enables modular controller synthesis with formal guarantees.

6. Practical Impact and Comparative Performance

The safe-by-design receding-horizon nonlinear MPC approach:

• Connects the classical NMPC paradigm with control Lyapunov/barrier function analysis, unifying optimality, safety, and invariance properties in a receding-horizon context.
• Enables the replacement of high-dimensional OCPs by trivial-sized QPs, with vast gains in computational speed and real-time tractability.
• Matches or exceeds the safety and convergence performance of classical NMPC and RTI, while guaranteeing strong satisfaction of hard state/input constraints even under nonconvex safe sets.
• Is directly applicable to general constrained nonlinear systems, including those arising in aerospace, robotics, and energy domains (Olucak et al., 26 Sep 2025).

The methodology opens avenues for deploying constrained nonlinear feedback controllers in time-critical or resource-limited systems without sacrificing formal guarantees or closed-loop performance.