Model Predictive Control with Multiple Constraint Horizons

TL;DR

This work develops an HC-MPC framework where state constraints are split across two horizons: a control-invariant set $\mathcal{X}_1$ and a (possibly non-invariant) tail $\mathcal{X}_2$, enabling suboptimality analysis without terminal penalties. By leveraging Relaxed Dynamic Programming, the authors derive a horizon-sensitive upper bound $J_{\infty}^{N,\tilde{N}}(x)$ tied to an explicit bound parameter $\alpha$, and introduce a complementary lower bound based on $\omega$ (and a secondary rate $\nu$) to certify suboptimality from below. They provide constructive ways to estimate $\alpha$, $\omega$, $\delta$, and $\nu$ from online HC-MPC data, and demonstrate the framework on nonlinear and linear safety-critical systems, showing how horizon choices shape closed-loop performance. The results offer a practical design tool for trading estimation accuracy against computational cost and safety guarantees in horizon-heterogeneous MPC settings.

Abstract

In this work we propose a Model Predictive Control (MPC) formulation that splits constraints in two different types. Motivated by safety considerations, the first type of constraint enforces a control-invariant set, while the second type could represent a less restrictive constraint on the system state. This distinction enables closed-loop sub- optimality results for nonlinear MPC with heterogeneous state constraints (distinct constraints across open loop predicted states), and no terminal elements. Removing the non-invariant constraint recovers the partially constrained case. Beyond its theoretical interest, heterogeneous constrained MPC shows how constraint choices shape the system's closed loop. In the partially constrained case, adjusting the constraint horizon (how many predicted- state constraints are enforced) trades estimation accuracy for computational cost. Our analysis yields first, a sub- optimality upper-bound accounting for distinct constraint sets, their horizons and decay rates, that is tighter for short horizons than prior work. Second, to our knowledge, we give the first lower bound (beyond open-loop cost) on closed-loop sub-optimality. Together these bounds provide a powerful analysis framework, allowing designers to evaluate the effect of horizons in MPC sub-optimality. We demonstrate our results via simulations on nonlinear and linear safety-critical systems.

Figures (6)

• Figure 1: System \ref{['eq:system']} under MPC \ref{['eq:VN_Ntilde']} for $N=6$, $N-\tilde{N}=2$. Input $u_h(2|x_k)$ produces the last open loop state $x_h(3|x_k)$ in $\mathcal{X}_1$. Subsequent open loop states are subject to $\mathcal{X}_2$. Closed loop states are depicted by $x_k$ and $x_{k+1}$.
• Figure 2: Black: $V_{N}^{\tilde{N}}(x_k)$ is equal to the sum of running costs $l(\cdot,\cdot)$ until the state $x_s = x_h(N-\tilde{N}+1|x_k)$, (last one in $\mathcal{X}_1$ - vertical dotted line). From $x_s$, value function's "tail" is computed by $V_{\tilde{N}-1}(x_s)$. States are shown below the nodes. Blue: Double headed arrow shows $x_{h}(1|x_k)$ in black and red are the same state. Red: An upper-bound of $V_{N}^{\tilde{N}}(x_{k+1})$ is derived using $x_{k+1}=x_h(1|x_k)$, built by the sum of running costs $l(\cdot,\cdot)$ until the state $x_s = x_h(N-\tilde{N}+1|x_k)$, which is the penultimate state required to be in $\mathcal{X}_1$. As the starting state now is $x_h(1|x_k)$ and the constraint horizon associated with $\mathcal{X}_1$ is also $N-\tilde{N}$, the state $x_h(1|x_s)$ will be the last required to be in $\mathcal{X}_1$. The "tail" of the value function can be computed by $V_{\tilde{N}}^{\tilde{N}}(x_s)$. Dashed black/red arrows have identical costs and cancel in $V^{\tilde{N}}_N(x_k)-V^{\tilde{N}}_N(x_{k+1})$.
• Figure 3: Reference frames for UAV uavtikz.
• Figure 4: Trajectory of UAV 1 avoiding collision with UAV 2.
• Figure 5: $\alpha$'s and closed-loop cost for $N=10$ as a function of $N-\tilde{N}$.

Theorems & Definitions (20)

• Remark 1
• Lemma 1
• proof
• Lemma 2
• Proposition 1
• Lemma 3
• Lemma 4
• Theorem 1
• proof
• Proposition 2