Tube-Based Robust MPC: Theory and Extensions

• Tube-based RMPC is a robust control methodology that computes an invariant tube around a nominal trajectory to enforce hard constraints under bounded disturbances and uncertainties.
• It employs constraint tightening via Minkowski differences and supports various tube parameterizations, such as rigid, homothetic, and ellipsoidal, to balance conservativeness and computational cost.
• The approach guarantees recursive feasibility, robust constraint satisfaction, and input-to-state stability, making it effective for both linear and nonlinear systems.

Tube-based Robust Model Predictive Control (RMPC) is a methodology for enforcing hard constraints in systems subject to bounded disturbances and model uncertainties. Its core principle is to compute an invariant set-valued "tube" centered on a nominal trajectory, with a feedback law that ensures all possible state trajectories, regardless of disturbance sequence, remain inside this tube. Nominal constraints are tightened to account for potential deviations, ensuring recursive feasibility and robust constraint satisfaction.

1. Problem Formulation and Core Principles

Tube-based RMPC frameworks address discrete-time, possibly nonlinear, systems subjected to additive and/or parametric uncertainties: $$x_{k+1} = f(x_k, u_k) + w_k, \quad x_k \in X,\, u_k \in U,\, w_k \in W$$ Here $f$ is locally Lipschitz, $X, U$ are compact convex sets, and $W$ is a compact convex disturbance set. The objective is to minimize a cost functional (typically quadratic or economic) subject to robust feasibility: $$\sum_{i=0}^{N-1} \ell(x_{k+i|k}, u_{k+i|k}) + V_f(x_{k+N|k})$$ subject to the evolution of the nominal trajectory and robustified state and input constraints.

The central construct is an invariant tube $T_k = \{x_k^n + e : e \in E\}$, where $x_k^n$ is the nominal (disturbance-free) prediction and $E$ is a robust positively invariant (RPI) set for the error dynamics. The control law is composed of the optimal nominal action and an ancillary (possibly nonlinear) feedback: $u_k = u_k^n + k(x_k - x_k^n)$ The feedback $k(\cdot)$ and the set $E$ are designed such that for any sequence of disturbances and for all $e_k \in E$, the next error state remains in $E$. The tube must satisfy the forward-invariance property: $$f(x_k^n + e_k, u_k^n + k(e_k)) - f(x_k^n, u_k^n) + w_k \in E$$ for all $w_k \in W$.

2. Tube Construction and Constraint Tightening

Constraint satisfaction under all admissible disturbances is ensured by constraint tightening via the Minkowski difference: $$x_k^n \in X_{\text{tight}} = X \ominus E, \quad u_k^n \in U_{\text{tight}} = U \ominus K(E)$$ where $K(E)$ is the image of $E$ under the ancillary feedback. For nonlinear systems, the ancillary feedback is typically nonlinear and designed using Lyapunov-based or incrementally stabilizing feedback techniques (Sun et al., 2017, Köhler et al., 2019).

For linear systems with additive or norm-bounded multiplicative uncertainty, ellipsoidal or polytopic sets are used to represent the tube cross-section. Homothetic, ellipsoidal, or polytopic tubes can be used depending on computational needs and conservativeness requirements (Massera et al., 2020, Parsi et al., 2022).

The tube dynamics and inclusion conditions are constructed so that:

• The error set contracts under closed-loop error dynamics and disturbance, guaranteeing bounded error.
• The tightened nominal constraints ensure the real system trajectory always respects the original hard constraints.

3. Nominal MPC and Online Optimization

The RMPC algorithm solves, at each time step, a finite-horizon optimal control problem for the nominal model subject to tightened constraints: $\min_{\{u_{k|k}^n\}} J_N(x_k^n, U_k^n)$

$$\begin{aligned} & x_{k+i+1|k}^n = f(x_{k+i|k}^n, u_{k+i|k}^n), \ & (x_{k+i|k}^n, u_{k+i|k}^n) \in X_{\text{tight}} \times U_{\text{tight}}, \ & x_{k+N|k}^n \in X_f \subset X_{\text{tight}} \end{aligned}$$

The solution provides an open-loop nominal control and state sequence; only the first input, combined with the ancillary feedback, is applied: $$u_k = u_{k|k}^{n,\star} + k(x_k - x_{k|k}^{n,\star})$$

The online complexity is dominated by the (tightened) MPC optimization, which remains tractable due to precomputed or implicit tube construction and the use of fixed-structure feedback. For nonlinear systems, recent algorithms separate offline computation of the incremental Lyapunov functions and tube-shape parameters from online trajectory optimization, reducing real-time computational requirements (Köhler et al., 2019).

4. Recursive Feasibility and Robust Stability

Recursive feasibility is ensured via proper choice of terminal set $X_f$ and tube invariance. Under the tube-based paradigm:

• If the RMPC OCP is feasible at time zero, recursive feasibility holds for all subsequent time steps.
• Tube invariance and monotonicity of the cost under the closed-loop law guarantee that constraint tightening is never violated.
• Input-to-state stability (ISS) is achieved with the closed-loop state converging to a compact set, the size of which depends on the disturbance bound: $\|x_k\| \leq \beta(\|x_0\|, k) + \gamma(w_{\max})$

For economic cost functions or in the absence of terminal ingredients, additional structural conditions such as strict dissipativity and turnpike properties guarantee bounded average costs and convergence to a robust optimal steady state (Schwenkel et al., 2019).

5. Tube Parameterization: Homothetic, Ellipsoidal, and Polytopic Tubes

Various parameterizations are utilized to trade off conservativeness and computational tractability:

• Rigid/Fixed tubes: The feedback gain and tube cross-section are fixed offline. The nominal trajectory is optimized online under fixed tightenings (Sun et al., 2017, Wang et al., 2019).
• Homothetic tubes: The tube cross-section is a fixed convex set scaled by a positive scalar ($\alpha_k$) at each step, optimizing both tube scaling and nominal trajectory online (Massera et al., 2020, Saccani et al., 2023).
• Ellipsoidal tubes: The tube section is an ellipsoid parameterized by its center and scaling, efficiently handled by LMIs or SDPs (Parsi et al., 2022).
• Varying tubes/concentric containers: Tube cross-section and its scaling are decision variables, allowing more accurate representation of multiplicative disturbances (Han et al., 4 Dec 2024).

The tube parameterization impacts feasible set volume, conservativeness, and online computational cost. Varying-tube and concentric container approaches yield improved feasible regions and smaller QPs compared to classical homothetic tubes (Han et al., 4 Dec 2024).

6. Extensions and Algorithmic Variants

Tube-based RMPC has been extended along several axes:

• Nonlinear and output-feedback cases: Use of incremental Lyapunov functions, nonlinear ancillary feedback, and state-dependent tubes (Köhler et al., 2019).
• Adaptive and learning-based RMPC: Integration of set membership or Gaussian process disturbance models allows online adaptation of tube bounds, reducing conservativeness in the presence of model uncertainty or time-varying disturbances (Köhler et al., 2019, Kiani et al., 2023).
• System-level and parallel algorithms: System Level Parameterization allows joint online optimization over the tube feedback law, reducing conservatism versus fixed-gain approaches (Sieber et al., 2021). Parallel explicit tube MPC leverages problem structure for real-time feasibility in large-scale or embedded systems (Wang et al., 2019).
• Koopman operator and data-driven lifting: Tube-based MPC using Koopman linear predictors handles nonlinear plants in a linear, reduced-complexity MPC, with tubes reflecting the modeling errors (Zhang et al., 2021).

7. Theoretical Guarantees and Performance

The principal theoretical results for tube-based RMPC include:

• Recursive feasibility: Feasibility of the tightened nominal OCP implies feasibility for all subsequent times, given the invariance of the error tube (Sun et al., 2017, Massera et al., 2020).
• Robust constraint satisfaction: The true state/input always respect the original (hard) constraints regardless of the disturbance realization, provided that the tube invariance and constraint tightening assumptions hold.
• Input-to-state stability: Closed-loop solutions satisfy ISS bounds, with convergence to a disturbance-dependent compact set (Sun et al., 2017, Köhler et al., 2019).
• Performance bounds: For economic cost, turnpike theory enables average cost guarantees without explicit terminal constraints (Schwenkel et al., 2019).
• Reduced conservatism compared to min-max MPC: Tube-based RMPC offers tractable methods for systems where full min-max design is computationally prohibitive, while empirical results demonstrate feasible region and performance improvements from advanced tube parameterizations or learning-based tube adaptation (Massera et al., 2020, Parsi et al., 2022, Kiani et al., 2023, Han et al., 4 Dec 2024).

References to Key Contributions

Selected foundational and recent works providing technical bases and algorithmic variants include:

• "Robust MPC for tracking of nonholonomic robots with additive disturbances" (Sun et al., 2017)
• "Tube-based Guaranteed Cost Robust Model Predictive Control for Linear Systems Subject to Parametric Uncertainties" (Massera et al., 2020)
• "A computationally efficient robust model predictive control framework for uncertain nonlinear systems" (Köhler et al., 2019)
• "Robust Economic Model Predictive Control without Terminal Conditions" (Schwenkel et al., 2019)
• "A System Level Approach to Tube-based Model Predictive Control" (Sieber et al., 2021)
• "Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets" (Parsi et al., 2022)
• "Robust Tube-based Model Predictive Control with Koopman Operators--Extended Version" (Zhang et al., 2021)
• "Learning Robust Model Predictive Control for Voltage Control of Islanded Microgrid" (Kiani et al., 2023)
• "Robust Model Predictive Control for Constrained Uncertain Systems Based on Concentric Container and Varying Tube" (Han et al., 4 Dec 2024)