System Level Tube-MPC

• SLTMPC is a robust control approach that jointly optimizes the nominal trajectory and error tube, offering improved performance over traditional tube-MPC.
• It is based on system-level parameterization, recasting robust MPC problems as convex optimization over closed-loop responses.
• SLTMPC reduces conservatism and computational complexity, making it effective for LTI and LTV systems facing additive and parametric uncertainties.

System Level Tube Model Predictive Control (SLTMPC) refers to a class of robust model predictive control (MPC) schemes that derive from system level parameterization (SLP) or system level synthesis (SLS). These methods recast robust constrained MPC problems—considering both additive disturbances and parametric/model uncertainties—as convex optimization over system response (closed-loop map) variables, enabling joint, often online, optimization of both the nominal trajectory and the feedback controller or “tube” that accounts for uncertainty. SLTMPC significantly reduces conservatism and computational complexity compared to classical tube-based approaches, provides transparent robustness guarantees, and, through targeted variants, is applicable to both LTI and LTV systems with polytopic or norm-bounded uncertainty (Chen et al., 2019, Sieber et al., 2021, Sieber et al., 2021, Sieber et al., 2024).

1. Theoretical Foundations and Motivation

Classical tube-based MPC (tube-MPC) maintains constraint satisfaction in the presence of uncertainty by evolving a nominal (planned) trajectory alongside an error “tube,” precomputed offline using a fixed feedback gain (e.g., LQR). The error dynamics are shaped by a fixed tube-law, and the controller enforces tightened constraints that account for worst-case disturbances and uncertainties, often yielding a conservative feasible region (Sieber et al., 2021). Tube-MPC with offline tubes suffers from:

• Fixed tube shaping: Feedback gain $K$ is predetermined and cannot be retuned online.
• Conservative constraint tightening: Constraint sets are often excessively tightened, as they must accommodate the worst-case disturbance evolution across the full horizon.
• Lack of joint optimization: The nominal plan and feedback are decoupled, potentially leading to suboptimal performance.

SLTMPC eliminates these limitations by directly optimizing over system-level closed-loop responses (such as system responses $\Phi_x$, $\Phi_u$ or their affine/FIR/Toeplitz representations), exploiting the structural properties of the closed-loop system, and embedding robustification mechanisms (e.g., via SLS) in the MPC formulation. This enables:

• Online, joint optimization of both tube shape (feedback law) and nominal trajectory.
• Direct inclusion of model uncertainty (norm-bounded or polytopic) and disturbance sets in a unified framework.
• Tractability via finite-dimensional, convex quadratic (or conic) programming (Chen et al., 2019, Sieber et al., 2021, Sieber et al., 2021, Sieber et al., 2024).

2. System-Level Parameterization and SLP/SLS Approach

SLTMPC is built upon the system-level parameterization (SLP) or system-level synthesis (SLS) framework, which characterizes all stabilizing controllers or state-feedback policies over a finite horizon in terms of structured, block-lower-triangular system response operators $\Phi_x$, $\Phi_u$. For discrete-time LTI or LTV systems

$x_{k+1} = A_k x_k + B_k u_k + w_k,$

with possibly time-varying $(A_k, B_k)$ and additive perturbations $w_k \in \mathcal{W}$, and model uncertainties

$$A_k = \hat A_k - \Delta_A(k), \quad B_k = \hat B_k - \Delta_B(k),$$

the stacked closed-loop state and input trajectories under a causal policy admit

$$\begin{bmatrix} x \ u \end{bmatrix} = \begin{bmatrix} \Phi_x \ \Phi_u \end{bmatrix} w,$$

where $\Phi_x$0.

These system responses are uniquely characterized (for finite-horizon, internally stabilizing policies) by the block-lower-triangular, affine achievability constraint

$\Phi_x$1

where $\Phi_x$2 is the block-downshift operator (Chen et al., 2019, Sieber et al., 2021). All robust and constraint-viable feedback policies are parameterizable as such system responses.

3. SLTMPC Problem Formulations

The essential SLTMPC finite-horizon OCP jointly optimizes over the system responses (and possibly additional nominal variables), leading to a convex program:

$\Phi_x$3

The cost $\Phi_x$4 accumulates stage and terminal costs evaluated along the nominal/expected trajectory, with constraints robustified over all allowable disturbances and uncertainties.

Modern SLTMPC variants include:

• Affine/FIR-constrained SLP formulations: With affine system-level parameterization and finite impulse response (FIR) constraints, all future disturbances vanish after a fixed horizon, simplifying robust tube construction (Sieber et al., 2021).
• Terminal constraint and controller design: Only a positively invariant (PI) terminal set is required, often computed via online scaling or Minkowski sums (Sieber et al., 2024, Sieber et al., 2021).
• Filter-based and polytopic inclusion approaches: For plant/model uncertainties with polytopic descriptions, error and input tubes are constructed using auxiliary filtered disturbance sets and online-optimized overapproximations, yielding computationally efficient robustification (Sieber et al., 2024).
• Adaptive and receding horizon strategies: To mitigate conservatism, the horizon can be adaptively selected online based on feasibility and value function minimization (Chen et al., 2021).

4. Tube Construction and Robust Constraint Handling

SLTMPC explicitly characterizes both the nominal trajectory and the error tube shape. Tube cross-sections are determined by the structured system-response blocks:

• For each stage $\Phi_x$5, the set of possible deviations due to all disturbances and uncertainty is given by the reachability of the error system under the parameterized $\Phi_x$6.
• Tightened state/input constraints enforce that at every step, for all $\Phi_x$7 and allowable $\Phi_x$8, the closed-loop evolution remains within the set constraints (Chen et al., 2019, Sieber et al., 2024, Sieber et al., 2021).

In the case of FIR-constrained SLP (affine parameterization), the error tubes are computed as time-varying Minkowski sums of disturbance sets through the system responses and reach an invariant cross-section after the horizon $\Phi_x$9 (Sieber et al., 2021).

Homothetic or diagonal filter-based tubes are often used for computational simplicity when the auxiliary disturbance set $\Phi_u$0 is a hyper-rectangle (Sieber et al., 2024).

5. Algorithmic and Computational Features

SLTMPC is formulated as a finite-dimensional convex quadratic or conic program, admitting efficient numerical solution and improved scalability relative to classical tube-MPC. The defining computational advantages include:

• No polytope vertex enumeration: All robustification over disturbance and uncertainty sets is handled without explicit enumeration, in contrast to robust MPC with polytopic/vertex description (Chen et al., 2019, Chen et al., 2021).
• Decision variable scaling: For horizon $\Phi_u$1, variables scale as $\Phi_u$2 and constraints as $\Phi_u$3, with many variants (especially offline-tube approaches) reducing the online solve to $\Phi_u$4 (Chen et al., 2019, Sieber et al., 2021, Sieber et al., 2024).
• Asynchronous computation: To mitigate online computational burden, asynchronous SLTMPC splits the optimization into fast execution of the nominal controller and slow, background tube recalculation via a secondary process. Tube data and terminal sets are managed in a memory buffer, and convex combinations of multiple precomputed tubes are fused to guarantee recursive feasibility and robust stability (Sieber et al., 2022, Sieber et al., 2024).
• Efficient implementation: In practical benchmarks, full SLTMPC solves are 10–20x faster than classical robust MPC with full polytope enumeration, and asynchronous SLTMPC enables control rates $\Phi_u$5 Hz (Sieber et al., 2022, Sieber et al., 2024). The offline/primary QP reduces to only nominal trajectory and fusion-weight variables.

6. Theoretical Guarantees and Comparative Performance

All core SLTMPC formulations provide the following guarantees under standard MPC assumptions:

• Recursive feasibility: The closed-loop control sequence remains feasible for all time, even as tubes or terminal sets are adaptively selected or recomputed (Sieber et al., 2021, Sieber et al., 2022, Sieber et al., 2024).
• Input-to-state stability (ISS): The optimal value function serves as an ISS Lyapunov function over the feasible set, and the closed-loop system is ISS with respect to the joint disturbance and model uncertainty (Sieber et al., 2021, Chen et al., 2021, Sieber et al., 2024).
• Reduced conservatism: SLTMPC’s feasible set size matches or exceeds that of classical tube-MPC, uniformly outperforming constant-gain, offline-tube relaxations in both feasible region and closed-loop cost (Sieber et al., 2021, Chen et al., 2021, Chen et al., 2019, Sieber et al., 2024).
• Terminal controller and set design: The introduction of online-scaled or auxiliary robust positively invariant (RPI) sets in terminal constraints yields robust, receding-horizon feasibility for both additive and parametric uncertainty (Sieber et al., 2024).

In numerical evaluations on double integrator and VTOL examples, SLTMPC reaches up to 95% of the maximal robust controlled invariant set, closing the gap with dynamic programming-derived best possible regions, and with closed-loop cost within 7% of full horizon SLTMPC, but with orders-of-magnitude lower per-step computation (Sieber et al., 2024, Sieber et al., 2022, Sieber et al., 2021, Chen et al., 2021).

7. Extensions, Applications, and Related Work

SLTMPC has been extended to LTV systems, polytopic and norm-bounded uncertainties, and settings with only a positively invariant (PI) terminal set. Its structural features facilitate:

• Extensions to distributed and explicit MPC via sparsity constraints in the SLP (Sieber et al., 2021).
• Handling of nonlinear or linear parameter-varying (LPV) plants by local linearization and robustification (Sieber et al., 2021).
• Incorporation of asynchronous computation, memory fusion, and fallback logic in real-time embedded controllers (Sieber et al., 2022, Sieber et al., 2024).
• Comparative analyses demonstrate that SLTMPC unifies and generalizes classical tube-MPC, robust SLS-MPC, and disturbance-feedback MPC, offering a favorable trade-off of computational tractability and conservatism.

A distinguishing property of recent SLTMPC formulations is the online, joint optimization of both the nominal plan and the full tube family, with robustness directly parameterized in the decision variables, thus achieving nonconservative constraint satisfaction without excessive tightening or increased online complexity (Chen et al., 2019, Sieber et al., 2021, Sieber et al., 2024).