Remote Tube-Based MPC

• Remote tube-based MPC is a control method that distributes computation between a remote unit for nominal trajectory optimization and a local unit for tube-based disturbance rejection.
• The framework guarantees recursive feasibility, hard constraint satisfaction, and input-to-state stability even when facing communication delays, packet losses, and model uncertainties.
• Empirical results demonstrate low tracking errors and fast remote solution times, validating its application in industrial automation, smart vehicles, and cyber-physical systems.

Remote tube-based Model Predictive Control (MPC) frameworks generalize classical tube-based MPC by explicitly distributing the computation of nominal trajectories and disturbance-rejecting feedback between remote (cloud/edge/server) and local (plant-side, embedded) modules. These schemes guarantee recursive feasibility, hard constraint satisfaction, and input-to-state stability for systems with bounded model uncertainty and disturbances, with explicit handling of network-induced phenomena—including delays, packet losses, and asynchronous computation. The architecture is motivated by applications such as industrial automation, smart vehicles, and cyber-physical platforms where plant and controller may be geographically separated and subject to lossy or delayed communications (Umsonst et al., 2024, Sieber et al., 2022).

1. Architectural Decomposition and Core Principles

Remote tube-based MPC (abbreviated RT-MPC, Editor's term) separates controller functionality into distinct remote and local components:

• Remote side: Hosts the nominal model, executes the MPC optimization to generate open-loop reference trajectories and steady-state assignments, and estimates the nominal state based on available uplink information.
• Local side: Implements a “smart actuator” that selects which nominal inputs to apply amid downlink losses, maintains a local copy of the nominal model for prediction, and enforces ancillary tube-based feedback for disturbance rejection and constraint robustness.

Communication occurs via explicitly defined uplink (plant to server) and downlink (server to plant) packet transactions, each carrying state, trajectory, or control sequence information, with logic for handling packet dropouts and out-of-order arrivals (Umsonst et al., 2024).

2. Plant, Nominal Model, and Tube Feedback Synthesis

The underlying plant is modeled as a constrained system subject to additive bounded disturbances:

$$x(k+1) = A x(k) + B u(k) + w(k), \qquad x(k) \in X,\, u(k) \in U,\, w(k) \in W$$

A nominal plant model (with $w \equiv 0$) is simulated locally using

$$x_\text{nom}(k+1) = A x_\text{nom}(k) + B u_\text{nom}(k)$$

The deviation $e(k) = x(k) - x_\text{nom}(k)$ is regulated by an ancillary tube controller—a static gain $K$ chosen such that $A_\text{cl} = A - B K$ is Schur. The applied input at the plant is then

$u(k) = u_\text{nom}(k) - K e(k)$

This split ensures that the closed-loop state $x(k)$ stays within a robust positively invariant (RPI) "tube" around the nominal trajectory determined by properties of $A_\text{cl}$ and the disturbance set $W$ (Umsonst et al., 2024).

3. Robust Invariant Tubes and Constraint Tightening

For disturbance set $w \equiv 0$0 and tube gain $w \equiv 0$1, the minimal RPI tube is

$w \equiv 0$2

Nominal MPC constraints are tightened so that, for all $w \equiv 0$3:

$w \equiv 0$4

where $w \equiv 0$5 denotes the Pontryagin difference. This tightening ensures that the true state and input—obtained by adding feedback action around the nominal—always satisfy original hard constraints.

4. Remote MPC Optimization and Loss-Resilient Control Logic

At each time step, the remote controller solves

$w \equiv 0$6

Upon computation, the remote sends a packet with the nominal control sequence, steady-state pair $w \equiv 0$7, the corresponding offset, and index of the last successful uplink. The local controller's consistent actuator logic selects the most recently received input based on pointer and loss indicators, applying the open-loop sequence or reverting to steady-state control with tube feedback if necessary. The nominal state is propagated locally, and the full input is reconstructed according to disturbance rejection logic (Umsonst et al., 2024).

5. Asynchronous and Distributed Tube Computation

Advanced variants exploit asynchronous tube computation. The “primary” process (onboard at the plant) solves the nominal MPC with whatever tubes are available, while the “secondary” process (remote or edge) computes new, less conservative tubes using system-level parameterizations (SLP) and communicates these updates independently of the control rate. The onboard memory holds a convex combination of tube candidates, and regular updates preserve recursive feasibility even in the presence of severe communication delays (Sieber et al., 2022).

A summary of this dual-process logic is presented below:

Process	Location	Frequency	Role
Primary	Onboard (plant)	Fast (control)	Nominal MPC, tube blending
Secondary	Remote/cloud	Slow (compute)	Tube optimization, updates

This division enables real-time operation with rigorous guarantees, even if tube-update packets are delayed or dropped.

6. Theoretical Guarantees: Feasibility, Safety, and Stability

Remote tube-based MPC frameworks guarantee:

• Recursive feasibility: If feasible at initialization, constraints remain satisfied for all future time-steps, even under bounded disturbances and sporadic communication loss.
• Robust constraint satisfaction: State and input always respect hard polytopic bounds due to tube-based tightening.
• Tube-bounded tracking: The true plant state converges to a neighborhood (defined by the tube) of the reference trajectory or set-point, almost surely under mild communication assumptions (infinite absence of perpetual blackouts).
• Implementation robustness: The system remains stable and constraint-admissible even if some remote updates fail to arrive; local memory and tube blending (as in the asynchronous setting) prevent disruption to feasibility (Umsonst et al., 2024, Sieber et al., 2022).

7. Applications and Empirical Performance

RT-MPC schemes have been instantiated on both nonlinear and linearized systems, notably in cart-pole regulation and interaction control for underwater vehicle-manipulator systems (Umsonst et al., 2024, Nikou et al., 2018). In empirical tests:

• RT-MPC and its variants (with or without full state feedback) remained recursively feasible across high packet loss rates ($w \equiv 0$8 up to 0.9).
• Real-time solution times for the remote controller remained within 5–8 ms for horizon $w \equiv 0$9 on example systems.
• Average tracking error increased smoothly with loss probability; “tube-less” R-MPC became infeasible under disturbance and communication loss, whereas RT-MPC maintained constraint satisfaction and feasible tracking at all times.
• Nominal and real state trajectories remained within computed tubes; control inputs never exceeded hard actuator limits.

These results encompass both simulation and practical control contexts, with publicly available implementation code for reference (Umsonst et al., 2024).

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References:

• "Remote Tube-based MPC for Tracking Over Lossy Networks" (Umsonst et al., 2024)
• "Asynchronous Computation of Tube-based Model Predictive Control" (Sieber et al., 2022)
• "A Tube-based MPC Scheme for Interaction Control of Underwater Vehicle Manipulator Systems" (Nikou et al., 2018)