Convex Model Predictive Control

Updated 17 November 2025

Convex Model Predictive Control is defined by receding-horizon control laws obtained by solving convex optimization problems that guarantee global optimality at every step.
It integrates advanced system identification methods such as input convex neural networks to approximate complex nonlinear dynamics while preserving convexity.
Robust implementations using techniques like tube MPC and chance constraints deliver reliable performance across applications from robotics to energy management.

Convex Model Predictive Control (MPC) comprises a class of optimal control techniques wherein the system dynamics, cost function, and constraints are constructed or approximated such that the resulting finite-horizon optimization problem is convex at every sampling step. This architecture enables global optimality, real-time feasibility, robustness to disturbances, and tractable incorporation of modern learning-based system identification methods. Convex MPC formulations are now deployed in domains ranging from neural-model-based policy synthesis to multi-agent navigation, smart grid optimization, chemical process control, and robust mission guidance in space operations.

1. Foundations of Convex Model Predictive Control

The defining characteristic of convex MPC is that the receding-horizon control law at each step is obtained by solving a convex optimization problem—typically a quadratic program (QP), quadratically-constrained QP (QCQP), second-order-cone program (SOCP), or semidefinite program (SDP). A convex program's feasible set and objective function satisfy convexity (and, in practice, usually strong convexity), which guarantees that any local minimum is a global minimum, and that efficient, robust algorithms may be deployed.

Core components:

System Dynamics: Either linear or encoded through convex surrogate models (e.g., input-convex neural nets, polynomial system relaxations).
Cost Function: Quadratic (LQR/energy) or convex polynomial in the stage cost and terminal cost.
Constraints: Linear (affine) or convex (convex polyhedra, ellipsoids, convex relaxations via LMIs/SOC constraints).
Robust/Chance Requirements: In some cases, constraints are imposed probabilistically or with respect to bounded uncertainty sets, convexified via liftings, tube approximations, or moment-SOS relaxations.

A sampling procedure typically proceeds as:

Measure the current plant state.
Formulate the finite-horizon convex optimization problem (cost, dynamics, constraints).
Solve using a convex solver (e.g., interior-point, operator-splitting, ADMM).
Implement the first element of the control sequence.
Repeat at the next sample.

The convexity of the underlying program is essential for real-time deployment and for leveraging advances in learning-based model parameterization.

2. Model Identification and Convex Surrogate Dynamics

Recent advances allow complex, nonlinear system identification within convex MPC. Notably, Input Convex Neural Networks (ICNNs) and their recurrent extensions, Input Convex Recurrent Neural Networks (ICRNNs), provide layered representations $f_\theta(x,u)$ that are convex in the control inputs $u$ , achieved via:

Non-negative weight matrices through architectural design and non-negativity constraints ( $W_k \ge 0$ , $D_k \ge 0$ at every layer).
Convex, non-decreasing activation functions (ReLU/softplus).
Input expansion to include both $u$ and $-u$ for bilinear models.

For multi-step/time-dependent systems, the ICRNN stacks internal states and validates convexity of the entire input trajectory. Training is conducted by minimizing mean-squared error (MSE) or prediction cost functions, with convexity-enforcing constraints on weights. Projection or parameterization ensures weights remain non-negative after updates. The resulting ICNN/ICRNN models are then embedded into the MPC loop, yielding convex programs even for highly nonlinear or hybrid physical systems (Chen et al., 2018, Wang et al., 13 Aug 2024, Bünning et al., 2020).

For polynomial systems subject to stochastic disturbance, convex surrogate dynamics can be built via lifting to infinite-dimensional linear programs (LPs) over measures, then relaxing to a tractable hierarchy of moment-SOS semidefinite programs (Jasour et al., 2016).

3. Robustness, Chance Constraints, and Tube Approximations

Robust convex MPC addresses system/model uncertainties and disturbances by constructing convex outer approximations or restrictions to guarantee persistent constraint satisfaction:

Tube MPC: The predicted state is enclosed in a tube (ellipsoid/hyperrectangle) whose growth is controlled via fixed-point dynamics, auxiliary feedback, and disturbance envelopes. Constraints are tightened by taking the Pontryagin-difference of the nominal set and the uncertainty tube; dynamic radii are updated recursively via linear inequalities or second-order cone constraints (Wullt et al., 29 Aug 2025, Lee et al., 2020).
Convex Restriction: By recasting system propagation as a fixed-point operator and enforcing self-mapping of convex tubes, the control sequence is optimized for worst-case disturbance realizations, with recursion guaranteeing robust feasibility (Lee et al., 2020).
System Level Synthesis (SLS)-based Inner Approximation: Parametrize closed-loop responses in terms of operators $(\Phi_x,\Phi_u)$ ; impose inner approximation constraints on state/input sets via facet tightening, yielding a single convex QP with guaranteed recursive feasibility and input-to-state stability (Chen et al., 2021).
Chance-Constrained MPC: For polynomial systems, probabilistic safety constraints are lifted to convex constraints via measure-theoretic relaxations. The solution converges to the true, generally nonconvex chance-constrained MPC as the hierarchy order $r \to \infty$ (Jasour et al., 2016).

4. Collision Avoidance and Geometric Convexification

Obstacle and collision avoidance for multi-agent or robotic systems necessitates geometric constraint reformulation for convex tractability:

Convex Feasible Set (CFS) Method: Collision avoidance constraints between agents are convexified by linearization (first-order approximation of the signed-distance function) around reference trajectories, producing affine inequalities that define convex safe regions (Zhou et al., 2021).
Convex Polygon-Aware Approximations: Vehicle and obstacle polygons are modeled; original disjunctive (OR) collision avoidance between non-convex sets is reformulated as tractable conjugate constraints (MSDE: minimum signed distance to edges, SVM-margined separating hyperplanes) to avoid mixed-integer overhead (Kojima et al., 8 May 2025).
Safe Corridors via Signed Distance Functions: For manipulators, safe motion is assured by defining convex balls along the reference path using a learned signed configuration-space distance function (SCDF), forming a corridor into which the tube MPC's nominal trajectory must lie (Wullt et al., 29 Aug 2025).
Polyhedral/Orbitope Relaxations for SE(n) Systems: Vehicle kinematics on $\mathrm{SO(n)}$ are convexified using spectrahedral (LMI-based) orbitopes rather than chart/linearization, converting transcendental constraints into semidefinite or second-order cone constraints compatible with convex MPC (Huang et al., 2014).

5. Solver Strategies, Explicit MPC, and Computational Aspects

Convex MPC admits a rich suite of solver methods, each with complexity/performance trade-offs:

Projected Interior-Point, ADMM, Convex QP/SDP Solvers: For moderate-scale or high-precision requirements in energy management or linear MPC (East et al., 2019).
MIQP with Explicit Policy Search: For machine-learning-based control, precompute explicit solution maps via multi-parametric QP (mpQP); track regions of the state space with binary selection variables in real time, preserving convexity via use of ICNN-based models (Wang et al., 13 Aug 2024).
Prediction-Correction Newton Schemes: Future MPC problems (with shifted plant states) exhibit similar KKT structure; sensitivity-based prediction steps (with Hessian inversion) followed by minimal correction steps guarantee quadratic convergence and minimal computational load per sampling interval (Paternain et al., 2019).
Multi-Convex and Split-Bregman Formulations: Multi-convex constraint structures (e.g., occlusion-free tracking in robot navigation) can be reduced via block alternation or augmented Lagrangians to a series of convex QPs and closed-form updates, leading to extreme real-time feasibility even on embedded hardware (Masnavi et al., 2021).

Empirically, convex reformulations provide dramatic reductions in solve-time—typically by factors of $10^2$ – $10^3$ compared to analogous nonconvex methods—while maintaining or improving control performance (Chen et al., 2018, Morstyn et al., 2017, Wen et al., 3 Mar 2024, Wullt et al., 29 Aug 2025).

6. Application Domains and Experimental Outcomes

Convex MPC concepts have been demonstrated in:

MuJoCo Locomotion: ICNN-based convex MPC achieves 10% higher cumulative reward and 5× faster solution time than nonconvex model-based RL approaches using random-shooting+MLP (Chen et al., 2018).
Building HVAC Control: Input convex neural nets yield 15–20% energy reduction relative to linear RC models, maintaining temperature comfort at all times (Bünning et al., 2020).
Microgrid Battery Energy Storage: Linear d–q models coupled to convex QCQP power flows allow for inclusion of line losses, voltage/current constraints, and realistic battery efficiency curves, at 1/1000 the computational cost of comparable nonconvex methods (Morstyn et al., 2017).
Crowded Robot Navigation: RL-guided subgoals coupled to convex region-constrained QP produce real-time collision-free trajectories; success rates in dynamic crowds exceed those of elastic band planners (TEB) by 10–13 points, with 30–40% reduction in trajectory length and time (Wen et al., 3 Mar 2024).
Distributed Vehicle Coordination: CFS-based convexification yields scalable sub-millisecond QP solves per agent, explicit deadlock resolution via adaptive speed, and robustness to low-level tracking error (Zhou et al., 2021).
Space Mission Guidance: Linearized convex MPC steps on equinoctial coordinates, with split-Edelbaum references, deliver robust, phasing-aware debris rendezvous even under realistic thrust uncertainties and misthrust events, with subpercent tracking slack (Wijayatunga et al., 2023).

7. Limitations, Open Issues, and Future Directions

While convex MPC offers strong theoretical and computational advantages, there are recognized limitations:

Model Expressiveness Constraints: Forced convexity/non-decreasing activation in learning-based models may limit ability to capture true non-monotonic dynamics (e.g., radiative cooling in buildings (Bünning et al., 2020)).
Conservatism of Robust/Chance Approximations: Inner approximations (SLS, tube-based, moment-SOS) can be conservative, excluding feasible policies in favor of guaranteed safety (Chen et al., 2021).
Explicit MPC Scalability: MIQP region selection can become impractical when the parameter space is high-dimensional or region counts grow super-linearly (Wang et al., 13 Aug 2024).
MILP Overhead in Logical/Obstacle Constraints: Some geometric or hybrid systems still require mixed-integer overhead for exact separation or logic constraints, trading off computational speed for optimality (Huang et al., 2014, Kojima et al., 8 May 2025).

Anticipated research directions include joint learning-convexification architectures, more efficient solver integration (operator splitting, conic methods), extension to stochastic and hybrid systems, and reduction of conservatism via adaptive or iterative constraint tightening. The unification of convex MPC with real-time certificate-based safety assurance is a central theme in robotics and cyber-physical systems.

*Key references include: (Chen et al., 2018, Bünning et al., 2020, Jasour et al., 2016, Wen et al., 3 Mar 2024, Zhou et al., 2021, Kojima et al., 8 May 2025, Wullt et al., 29 Aug 2025, Morstyn et al., 2017, Wang et al., 13 Aug 2024, Lee et al., 2020, Huang et al., 2014, Paternain et al., 2019, Wijayatunga et al., 2023, Masnavi et al., 2021, Bemporad et al., 2015, Lishkova et al., 2023, East et al., 2019, Chen et al., 2021).