Robust Economic Tube MPC

• The paper introduces a novel tube-based formulation that explicitly handles uncertainty via polytopic linear differential inclusions to ensure robust constraint satisfaction.
• It formulates periodic economic objectives into a convex quadratic program, guaranteeing recursive feasibility and convergence to optimal trajectories under disturbances.
• Practical applications in chemical and energy systems demonstrate scalability, computational efficiency, and robust performance guarantees.

Robust Economic Tube Model Predictive Control (MPC) is an advanced framework for optimizing the economic performance of uncertain dynamical systems while ensuring robust state and input constraint satisfaction. It extends standard economic MPC by introducing tube-based methods to explicitly manage uncertainties, particularly in systems with additive disturbances and model uncertainty, and is distinguished by its ability to rigorously guarantee robustness and feasibility in the face of system and environment variability. Recent developments, especially in the context of periodic operation and polytopic system descriptions, provide tractable optimization formulations and concrete performance guarantees.

1. Problem Structure and Mathematical Formulation

Robust Economic Tube MPC targets systems characterized by uncertainty modeled as polytopic linear differential inclusions (LDIs), where system matrices $(A,B)$ evolve within the convex hull of a finite set of mode pairs, and disturbances $w \in \mathcal{W}$ are bounded, convex, and compact. The prediction model takes the form: $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$ State and input constraints are posed as $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$ and $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$, with $\mathcal{X}$ and $\mathcal{U}$ closed and convex.

The economic objective is periodic: for period $T$, the stage cost $\ell_{[t+j]}(z_j, v_j)$ (possibly time-varying or weighted by exogenous signals) yields a one-period cost

$$M_t(z, v) = \sum_{j=0}^{T-1} \ell_{[t+j]}(z_j, v_j)$$

This framework is motivated by the observation that periodic trajectories are often optimal under economic objectives in many process systems subject to exogenous variability (Badalamenti et al., 3 Dec 2025).

2. Tube Representation and Invariance Constraints

Trajectory tubes are parametrized using polytopic cross-sections: $w \in \mathcal{W}$0 Configuration cones $w \in \mathcal{W}$1 constrain $w \in \mathcal{W}$2 to preserve the combinatorial structure (facet–vertex incidence) of the polytope. Vertex representations $w \in \mathcal{W}$3 provide efficient evaluation of invariance and constraint satisfaction at all tube vertices.

System invariance is encoded via a set $w \in \mathcal{W}$4 containing all valid local tube transitions: $w \in \mathcal{W}$5 where $w \in \mathcal{W}$6. If every $w \in \mathcal{W}$7 is in $w \in \mathcal{W}$8, then the sequence $w \in \mathcal{W}$9 forms a robust forward invariant tube (RFIT), providing robust constraint satisfaction for all admissible uncertainty realizations (Badalamenti et al., 3 Dec 2025).

3. Optimization-Based Control Synthesis

The closed-loop control problem is recast as a convex program: $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$0 subject to:

• $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$1 (initial condition covers the current state)
• $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$2 for $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$3 (tube invariance)
• $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$4 for $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$5 (periodic tube invariance)
• $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$6 (periodicity enforcement)
• $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$7 (terminal set)

Decision variables encode both the finite-horizon tube to be tracked $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$8 and an artificial periodic tube $$x_{k+1} = \sum_{i=1}^M \lambda_i (A_i x_k + B_i u_k) + w_k, \qquad \lambda_i \geq 0,\; \sum \lambda_i = 1,\; w_k \in \mathcal{W}$$9, with artificial variables supporting recursive feasibility and robust constraint satisfaction under possible online changes of the economic criterion. Robustification is tightly encoded: invariance constraints in $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$0 propagate all tube vertices, across all polytopic models, and under maximal disturbances (Badalamenti et al., 3 Dec 2025).

4. Theoretical Guarantees: Feasibility, Convergence, and Performance

The robust tube MPC framework guarantees:

• Recursive feasibility and robust constraint satisfaction for admissible initializations, as invariance is explicitly enforced over all possible system realizations.
• Convergence to the optimal periodic tube when the economic criterion remains constant, via dynamic real-time optimization of the periodic tube variables and a Lyapunov argument applied to the cost function value gap. The Lyapunov candidate $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$1 decreases along closed-loop executions and converges to zero as $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$2 (Badalamenti et al., 3 Dec 2025).
• The cost incurred closely aligns with the optimum over finite and infinite time horizons, up to vanishing terms as the prediction horizon increases, mirroring bounds previously established for nominal economic MPC (Klöppelt et al., 2021).

Terminal ingredients are constructed as follows: the terminal cost $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$3 and corresponding terminal set $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$4 provide a recursively invariant region, and the Lyapunov-type decrease

$x \in \mathcal{X} \subset \mathbb{R}^{n_x}$5

establishes contractivity and stability under appropriate weighting (matrices $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$6, $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$7 and $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$8) (Badalamenti et al., 3 Dec 2025).

The same theoretical structure underpins older tube-based economic MPC schemes, both with and without terminal conditions (Klöppelt et al., 2021, Schwenkel et al., 2019). In these, robust optimal steady states (ROSS) are characterized, and strict dissipativity and turnpike properties yield that the average closed-loop cost approaches the ROSS optimum for long horizons, modulo vanishing residual terms.

5. Computation, QP Reformulation, and Scalability

To enhance computational tractability, a quadratic surrogate for the periodic economic cost is constructed. Assuming the Lipschitz continuity of $x \in \mathcal{X} \subset \mathbb{R}^{n_x}$9 (with constant $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$0),

$u \in \mathcal{U} \subset \mathbb{R}^{n_u}$1

satisfies $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$2. All constraints remain linear, allowing the robust economic tube MPC to be implemented as a quadratic program (QP) (Badalamenti et al., 3 Dec 2025). Algorithmically, the QP is solved at each time step with warm-starts based on cyclic shifts, convex-combination control action construction, and real-time updates to the linearization point.

Complexity scales linearly with both the prediction/warm start horizon $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$3 and the periodic tube horizon $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$4, with a variable count $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$5, and the number of invariance constraints per step scaling proportional to $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$6. Reported solve-time reductions by over an order of magnitude are possible by leveraging warm-starts, offline PFIT computation, and efficient QP solvers.

6. Connections to General Tube-Based Robust Economic MPC

The configuration-constrained periodic tube approach generalizes and enhances classic tube-based robust economic MPC, as detailed for both terminal-included (Klöppelt et al., 2021) and terminal-free (Schwenkel et al., 2019) formulations:

• The fundamental design consists of a nominal MPC with constraint tightening based on a robust control invariant (RCI) tube $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$7, state-feedback law $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$8, and robust stage cost selection. Real states are confined to $u \in \mathcal{U} \subset \mathbb{R}^{n_u}$9, and constraints on $\mathcal{X}$0 are pre-tightened.
• Turnpike properties and dissipativity ensure that nominal trajectories spend most of their time near the ROSS, yielding tight bounds on the closed-loop average cost.
• Both traditional (single steady-state) and configuration-constrained (periodic) robust economic tube MPC retain feasibility, stability (as ISS in the set sense), and robust performance as prediction horizons grow.

The configuration-constrained formulation uniquely addresses the periodicity of the economic objective, enables efficient high-dimensional polytope parameterization, and ensures exact satisfaction of invariance by encoding all uncertainty as a single consistent convex hull condition (Badalamenti et al., 3 Dec 2025).

7. Applications and Practical Considerations

Robust economic tube MPC is directly applicable to industrial settings requiring periodic economic operation under uncertainty, including chemical process systems, energy systems, and robotics. Empirical demonstrations include a benchmark ball-plate system with eight states (Badalamenti et al., 3 Dec 2025). Implementation success depends critically on:

• The ability to compute sufficiently small RCI tubes $\mathcal{X}$1 to minimize conservatism,
• Selection of stage costs reflecting worst-case or averaged tube behavior,
• Proper horizon selection to balance computational effort and convergence rate,
• and, in periodic cases, careful configuration cone design to preserve polytope structure during optimization.

The method delivers a single-layer robust economic MPC that unifies constraint tightening, tube invariance, and periodic economic objectives, with theoretical support for recursive feasibility, robust constraint satisfaction, and asymptotic convergence. Quadratic-program reformulations enable real-time deployment with scalability to moderate-sized systems.

References: (Badalamenti et al., 3 Dec 2025, Klöppelt et al., 2021, Schwenkel et al., 2019)