Chance-Constrained MPC

• Chance-Constrained MPC is an advanced control method that combines finite-horizon optimal control with probabilistic constraint enforcement for uncertain discrete-time systems.
• It employs constraint tightening and probabilistic tubes to guarantee that state and input constraints are met with prescribed violation probabilities.
• The approach ensures recursive feasibility and closed-loop stability by solving a strictly convex quadratic program while accounting for unbounded disturbances.

Chance-Constrained Model Predictive Control (CC-MPC) is an advanced control methodology for discrete-time systems under uncertainty, combining finite-horizon optimal control with probabilistic guarantees on constraint satisfaction. CC-MPC explicitly models system disturbances or uncertainty, enforcing that specified state or input constraints are satisfied with a prescribed probability, rather than deterministically, at each control step. This probabilistic relaxation enables reduced conservatism and greater operational efficiency compared to robust MPC, particularly when disturbances are unbounded or only partial knowledge of their statistics is available.

1. Formulation and Disturbance Modeling

The canonical CC-MPC problem assumes a discrete-time linear system

$x_{k+1} = A x_k + B u_k + w_k,$

where $x_k \in \mathbb{R}^n$ is the state, $u_k \in \mathbb{R}^m$ the control input, and $w_k$ is an additive disturbance modeled as an i.i.d. sequence with prescribed mean $\mu_\omega$ and covariance $\Sigma_\omega$, but possibly unbounded or non-Gaussian. The novelty in recent formulations is that only the first and second moments $(\mu_\omega, \Sigma_\omega)$ are assumed known, eschewing any requirement on boundedness or a specific distributional form (Wang et al., 19 Sep 2024).

Chance constraints are imposed on given polytopic sets $X = \{ x: Hx \le h \}$ and $U = \{ u: Lu \le h_u \}$, requiring: $$\Pr\{ x_k \in X \} \ge 1-\epsilon_x,\quad \Pr\{ u_k \in U \} \ge 1-\epsilon_u$$ for all $k$, where $\epsilon_x, \epsilon_u \in (0,1)$ are the allowable violation probabilities.

2. Constraint Tightening and Probabilistic Tubes

To guarantee chance constraint satisfaction, CC-MPC employs a decomposition: $x_k = z_k + s_k,\quad u_k = v_k + K s_k$ where $(z_k, v_k)$ are the nominal (open-loop) trajectories and $s_k$ is the error due to disturbances, attenuated via a feedback gain $K$. The feedback law is selected so that $A_K = A + BK$ is Schur.

Original state and input polytopes are replaced by "tightened" sets: $Z = X \ominus R_x,\qquad V = U \ominus KR_u,$ where $R_x, R_u$ are polytopic probabilistic tubes computed to guarantee: $$\Pr(s_k \in R_x) \ge 1-\epsilon_x, \quad \Pr(K s_k \in K R_u) \ge 1-\epsilon_u.$$ The minimal robust positively invariant (mRPI) set for the error dynamics

$s_{k+1} = A_K s_k + w_k$

is constructed via polytopic representations and support function evaluations on ellipsoidal confidence regions derived from $(\mu_\omega, \Sigma_\omega)$. A Chebyshev-type bound is employed to ensure probabilistic positive invariance (PPI): $$E(\Sigma_k, \mu_k, n/\epsilon) \subseteq R^\epsilon_k,\quad R^\epsilon_{k+1} = A_K R^\epsilon_k \oplus E(\Sigma_\omega, \mu_\omega, n/\epsilon).$$ If $R(q^*)$ satisfies

$$A_K R(q^*) \oplus E(\Sigma_\omega, \mu_\omega, n/\epsilon) \subseteq R(q^*),$$

then it guarantees the prescribed probability level (Wang et al., 19 Sep 2024).

3. Optimization Problem and Theoretical Guarantees

At each control step, the CC-MPC optimization solves a strictly convex quadratic program (QP)

$$\begin{aligned} \min_{\{z_{t|k}, v_{t|k}\}}&\; \sum_{t=0}^{N-1} \left( z_{t|k}^\top Q z_{t|k} + v_{t|k}^\top R v_{t|k} \right) + z_{N|k}^\top P z_{N|k} \ \text{s.t.}\quad & z_{0|k} = z_k,\; z_{t+1|k} = A z_{t|k} + B v_{t|k},\; \forall t \ & H z_{t|k} \le h - \Delta^x,\; L v_{t|k} \le h_u - \Delta^u,\; \forall t \ & z_{N|k} \in Z_f \end{aligned}$$

with deterministic tightened polytopes. The tightenings $\Delta^x_i, \Delta^u_j$ are given by the support functions of $R_x$ and $K R_u$ against the constraint normal vectors. The terminal set $Z_f$ is constructed as the maximal invariant set for the local feedback law, and the terminal cost is based on the discrete Lyapunov function for the LQR design.

Recursive feasibility and closed-loop probabilistic stability are ensured under simple conditions (stabilizability, interiority, and the Lyapunov decrease condition):

• The optimization problem remains feasible for all $k$.
• The nominal closed-loop system converges to zero.
• The true state $x_k$ converges in distribution to the stationary process for the error dynamics, with chance constraints holding at every step (Wang et al., 19 Sep 2024).

4. Algorithmic Steps and Computational Complexity

Implementation requires:

• Offline: Selection of feedback gains $K$, $K_f$; computation of Lyapunov matrix $P$; calculation and assembly of the polytopes $R_x, R_u$, tightened sets, and terminal set $Z_f$.
• Online: At each sample, form the nominal state $z_k$ (using $x_k-s_{k-1}$ or shifted sequence), solve the QP for $(z_{t|k}, v_{t|k})$, and apply the feedback law $u_k = v_{0|k}^* + K(x_k-z_k)$.

The offline cost is dominated by a single small LP and polytope differences. Online complexity matches nominal tube MPC, i.e., one QP of size $\mathcal{O}(N(n+m))$ per step (Wang et al., 19 Sep 2024).

5. Generalizations and Related CC-MPC Schemes

Chance-constrained tube MPC as in (Wang et al., 19 Sep 2024) is part of a broader class of approaches leveraging probabilistic invariance, constraint tightening, and separation of mean and variance effects:

• Probabilistic reachable sets under arbitrary convex uncertainty (including non-Gaussian or only moment-based models) (Hewing et al., 2018, Mark et al., 2021).
• Multi-agent and distributed CC-MPC via region overlap and Chebyshev-type bounds for collision avoidance (Lyons et al., 2011).
• Nonlinear and data-driven CC-MPC using scenario sampling, moment relaxations, or online regression for constraint tightening parameters (Capone et al., 2023, Jasour et al., 2016, Streif et al., 2014).
• Distributionally robust variants encompassing ambiguity in uncertainty models, with recursive feasibility established by updating ambiguity sets online (Schuurmans et al., 2020).
• Discounted and infinite-horizon probabilistic constraints where the sum of discounted violation probabilities is bounded, eliminating the need for bounded disturbances or invariant terminal sets (Yan et al., 2020, Yan et al., 2018).
• Adaptive relaxation and online learning schemes adjusting constraint margins based on empirical violation rates, improving closed-loop tightness and reducing conservatism (Ghosh et al., 4 Jun 2024).
• Extensions to multi-modal uncertainty representations, e.g., Gaussian mixture models for moving obstacles (Ren et al., 8 Jan 2024).

6. Practical Significance and Application Domains

Chance-constrained MPC is widely applied in domains where strict deterministic constraint satisfaction is infeasible or economically suboptimal, and probabilistic safety margins are operationally justified. Notable applications include:

• Urban drainage and water networks under load forecast uncertainty, yielding improved overflow management and robustness compared to deterministic MPC, while retaining tractable QP/SOCP complexity (Svensen et al., 2020, Svensen et al., 2020, Svensen et al., 2020).
• Robotic and autonomous systems (e.g., quadrupedal locomotion, multi-agent planning, collision avoidance), using CC-MPC to achieve high performance with quantifiable risk (Trivedi et al., 5 Nov 2024, Li et al., 2023, Lyons et al., 2011).
• Spacecraft rendezvous in nonlinear or time-varying dynamics—probabilistic constraint tightening enables robust performance under high disturbance uncertainty in cislunar environments (Sanchez et al., 13 Jan 2025).
• Grid operation, BESS dispatch, and energy systems, where nonconservative chance constraint approaches deliver economic benefits with explicit violation control (Ghosh et al., 4 Jun 2024).

7. Limitations, Conservatism, and Current Research Directions

Main limitations include:

• Conservatism introduced by the Chebyshev inequality or union-bounding, especially when only moments are used.
• Computational complexity with increasing system size or horizon in sample-based and scenario-driven methods.
• The challenge of recursive feasibility under nonstationary, distributionally ambiguous, or multimodal disturbance models.

Recent research seeks less conservative margin computation (e.g., analytic or online regression-based tightening), tighter management of empirical violation rates, learning-based ambiguity set tightening, and distributed CC-MPC for large-scale, networked, or uncertain systems (Capone et al., 2023, Ghosh et al., 4 Jun 2024, Schuurmans et al., 2020, Trivedi et al., 5 Nov 2024). In summary, the minimal-moment, polytopic invariant set approach (Wang et al., 19 Sep 2024) represents a scalable, theoretically well-founded solution for certifiably safe, real-time MPC with chance constraints under broad stochastic uncertainty.