---
title: 'Tube-Based Robust MPC: Theory and Extensions'
url: https://www.emergentmind.com/topics/tube-based-robust-model-predictive-control-rmpc
type: topic
---

# Tube-Based Robust MPC: Theory and Extensions

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 [1703.03101, 1910.12081].

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 [2012.05349, 2204.02134].

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 [1910.12081].

## 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 [1911.12235].

## 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 [1703.03101, 1910.03271].
- **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 [2012.05349, 2309.06591].
- **Ellipsoidal tubes:** The tube section is an ellipsoid parameterized by its center and scaling, efficiently handled by LMIs or SDPs [2204.02134].
- **Varying tubes/concentric containers:** Tube cross-section and its scaling are decision variables, allowing more accurate representation of multiplicative disturbances [2412.02985].

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 [2412.02985].

## 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 [1910.12081].
- **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 [1911.02899, 2309.00742].
- **System-level and parallel algorithms:** System Level Parameterization allows joint online optimization over the tube feedback law, reducing conservatism versus fixed-gain approaches [2103.02460]. Parallel explicit tube MPC leverages problem structure for real-time feasibility in large-scale or embedded systems [1910.03271].
- **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 [2108.13011].

## 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 [1703.03101, 2012.05349].
- **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 [1703.03101, 1910.12081].
- **Performance bounds:** For economic cost, turnpike theory enables average cost guarantees without explicit terminal constraints [1911.12235].
- **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 [2012.05349, 2204.02134, 2309.00742, 2412.02985].

## 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" [1703.03101]
- "Tube-based Guaranteed Cost Robust Model Predictive Control for Linear Systems Subject to Parametric Uncertainties" [2012.05349]
- "A computationally efficient robust model predictive control framework for uncertain nonlinear systems" [1910.12081]
- "Robust Economic Model Predictive Control without Terminal Conditions" [1911.12235]
- "A System Level Approach to Tube-based Model Predictive Control" [2103.02460]
- "Scalable tube model predictive control of uncertain linear systems using ellipsoidal sets" [2204.02134]
- "Robust Tube-based Model Predictive Control with Koopman Operators--Extended Version" [2108.13011]
- "Learning Robust Model Predictive Control for Voltage Control of Islanded Microgrid" [2309.00742]
- "Robust Model Predictive Control for Constrained Uncertain Systems Based on Concentric Container and Varying Tube" [2412.02985]

Source: https://www.emergentmind.com/topics/tube-based-robust-model-predictive-control-rmpc