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Configuration-Constrained Tube MPC for Difference-of-Convex Nonlinear Systems
Filippo Badalamenti, Jose A. Borja-Conde, Filiberto Fele, Teodoro Alamo, Daniel Limon
TL;DR
Nonlinear robust MPC must propagate sets through dynamics whose nonlinear images are difficult to characterize tractably. The paper combines DC-based directional bounds with configuration-constrained polytopic tubes in a convex controller. In the numerical example, predicted tubes contain the closed-loop state and contract toward the target, while 10- or 12-vertex templates offer the reported favorable complexity trade-off.
Problem
Nonlinear images of convex sets generally lack finite boundary characterizations, making tight robust successor-set computation difficult; the set-valued nonlinear robust MPC problem is generally intractable.
Source: §1 · p. 1, §2.2 · p. 4Method
Results
The predicted tubes contain the closed-loop state and contract toward the target set; in the example, templates with 10 or 12 vertices provide the most favorable trade-off.
Source: §5 · p. 8, §5.1 · p. 9Takeaways & Limitations
An implicit terminal condition guarantees recursive feasibility and convergence of the predicted tube to a target robust control invariant set.
Source: §6 · p. 9The finite-dimensional propagation certificate is sufficient but not necessary, so its vertex-input restriction may add conservatism beyond the DC outer approximation.
Source: §3.1 · p. 5
Abstract
from arXiv · showhide
This paper develops a convex tube model predictive control formulation for constrained nonlinear systems. We consider dynamics described by a discrete-time state-space model with parametric and additive uncertainty that admits a difference-of-convex decomposition. Convex directional bounds of the nonlinear dynamics are combined with configuration-constrained polytopic tubes, whose predefined combinatorial structure yields an affine parameterization of their vertices. The resulting finite-dimensional sufficient conditions certify robust one-step tube propagation under parametric uncertainty and additive disturbances, while allowing the tube geometry and an associated vertex control law to be optimized jointly in a single convex program. An implicit terminal condition guarantees recursive feasibility and convergence of the predicted tube to a target robust control invariant set. Numerical results illustrate the closed-loop properties and the trade-off between geometric flexibility and computational complexity.
1 Introduction
Robust tube MPC handles uncertainty through set-valued predictions, but nonlinear propagation makes tight successor sets difficult to characterize tractably. This paper combines DC-based propagation bounds with configuration-constrained tubes to optimize nonlinear tube geometry and vertex inputs jointly.
1 Introduction
- Tube MPC predicts sets enclosing all trajectories generated by admissible uncertainty, rather than a single state trajectory.For linear systems and linear difference inclusions, this set-based description leads to tractable formulations with recursive feasibility and stability guarantees.Source: §1 · p. 1
- Nonlinear images of convex sets generally lack finite boundary characterizations, making tight robust successor computation difficult and motivating conservative propagation bounds.Existing approaches trade off modeling assumptions, computational complexity, and conservatism; DC decompositions offer a structure for constructing tractable bounds.Source: §1 · p. 1
- Prior DC-based bounds convexify nonlinear propagation, but their combination with online-optimized flexible tube geometry remained open.Configuration-constrained tubes provide affine vertex parameterizations and can reduce conservatism relative to more restrictive tube parameterizations, at adjustable computational cost.Source: §1 · p. 2
- The proposed method combines convex directional bounds with configuration-constrained polytopes to certify robust one-step propagation under parametric uncertainty and additive disturbances.The resulting controller jointly optimizes tube geometry and vertex inputs in one convex program; an implicit terminal condition guarantees recursive feasibility and convergence to a target robust control invariant set.Source: §1 · p. 2
2 Problem Formulation
The section formulates uncertain nonlinear dynamics and a set-based robust MPC problem, then introduces convex bounds and structured polytopes to make tube propagation finitely representable. Its setup relies on a DC decomposition and configuration-constrained tube sections.
2.1 Uncertain Nonlinear Systems
- The dynamics include parametric uncertainty θ_t ∈ Θ and additive disturbances w_t ∈ W, each varying arbitrarily within its set; the model assumes compact, convex constraint and uncertainty sets containing the origin.The DC assumption writes each component of f as a difference of finite-valued convex functions, though the chosen decomposition affects convexification and conservatism.Source: §2.1 · p. 2
- DC component bounds yield a convex upper bound on each directional projection of the dynamics over the domain.The construction uses affine minorants at a reference point; the point and any direction-dependent decompositions are selected offline and fixed online.Source: §2.1 · pp. 2–3
2.2 Robust Control Tubes
- A robust control tube is a sequence of sets whose next section contains the one-step successors of the current section for admissible controls and all uncertainties.A robust control invariant set is the special case of a tube with the same set at both steps.Source: §2.2 · p. 3
- The canonical finite-horizon set-valued MPC problem constrains each section by robust propagation, state constraints, and a terminal set, but is generally intractable.Its decision variables are sets and its propagation condition involves nonlinear robust reachability.Source: §2.2 · pp. 3–4
- The tractable approach replaces nonlinear dynamics with a convex outer approximation and restricts tube sections to polytopes with fixed combinatorial structure.Together these choices provide a sufficient one-step propagation certificate encoded in finite-dimensional form.Source: §2.2 · p. 4
2.3 Configuration-Constrained Polytopes
- Tube sections use a fixed template matrix F and manipulable offsets y, with admissible offsets preserving the polytope’s face configuration.This restriction keeps facet and vertex positions affine in y; configuration triples encode the family, with construction and flexibility-versus-complexity trade-offs discussed in.Source: §2.3 · p. 4
- Constructing a configuration triple is nontrivial, and the paper describes an offline nonlinear-programming procedure to initialize templates through sign-preserving deformation.The procedure extends an existing initialization approach from linear difference inclusions to the DC setting.Source: §2.3 · p. 4
3 Convex Reformulation for Tube Propagation
Facet-direction convex bounds turn nonlinear dynamics into finite robust-propagation inequalities; configuration-constrained vertices yield a sufficient tube certificate and an invariant-set condition.
3 Convex Reformulation for Tube Propagation
- Convex componentwise upper bounds are constructed along facet normals at a fixed design-time linearization point.The bounds apply over the state, input, and parameter sets.Source: §3 · p. 4
- The disturbance support vector and finite inequalities (14) provide a sufficient certificate for robust one-step propagation.The support vector measures the disturbance set along the facet-normal directions.Source: §3 · pp. 4–5
3.1 Configuration-Constrained RCTs
- The configuration-constrained formulation imposes propagation inequalities at polytope vertices using a collection of admissible vertex inputs.Convex combinations of those inputs provide admissible state-dependent inputs across the tube.Source: §3.1 · p. 5
- Membership in the set eS guarantees robust propagation, but the vertex-input condition can add conservatism beyond the DC outer approximation.The converse need not hold when a state-dependent admissible input cannot be expressed as a convex combination of admissible vertex inputs.Source: §3.1 · p. 5
- A self-loop triple (y, u, y) in eS certifies the configuration-constrained polytope X(y) as a robust control invariant set.This follows by applying the one-step propagation condition to the same set.Source: §3.1 · p. 5
4 Nonlinear CC-TMPC for Regulation
The nonlinear CC-TMPC controller jointly optimizes tube geometry and vertex inputs toward a fixed robust control invariant target. Its implicit terminal condition supports recursive feasibility and convergence of the predicted tube to that target.
4 Nonlinear CC-TMPC for Regulation
- The MPC design optimizes tube parameters and vertex inputs over a prediction horizon toward a fixed robust control invariant target.A convex regulation cost and contractive one-step condition induce an implicit terminal set in tube-configuration space.Source: §4 · pp. 5–6
4.1 Implicit Terminal Set
- The implicit terminal set supports Lyapunov descent, with terminal cost T null at the target and satisfying descent on the set.The set is nonempty whenever the target robust invariant condition is feasible, for any contraction factor in [0, 1).Source: §4.1 · p. 6
4.2 Closed-Loop Control Law
- Any simplex-feasible barycentric representation can produce an admissible feedback input; the implementation selects one minimizing control effort with a QP.The shifted candidate sequence establishes recursive feasibility at the next time step.Source: §4.2 · p. 7
- Under the stated positive-definite weight and contraction conditions, the closed loop remains feasible and its tube parameters converge to the target pair.The optimized tube section containing the state also converges to the target robust control invariant set.Source: §4.2 · pp. 7–8
5 Numerical Example
The Duffing-oscillator example shows closed-loop tube containment and contraction toward a target invariant set. The computational study finds a trade-off between feasible-region size and online time as template vertex count increases.
5 Numerical Example
- The example uses a forward-Euler discretization of an uncertain Duffing oscillator with bounded parameters, additive disturbance, state constraints, and input constraints.The sampling time is 0.2 s; the uncertain parameter set is [0.8, 1.2] × [0.9, 1.1], and the disturbance set is [−0.25, 0.25].Source: §5 · p. 8
- Convex directional bounds use a DC decomposition of the uncertain dynamics, including a convexity constant chosen over the constrained domain for the cubic restoring force.The resulting discrete-time DC maps are formed from the continuous-time convex components.Source: §5 · p. 8
- The dodecagonal configuration-constrained family uses 12 facets and vertices, and its feasible-region approximation is built from 100 support directions.The target pair is computed using the stated convex criterion.Source: §5 · p. 8
- The optimized tube sections contain the closed-loop states and contract toward the target invariant set; the feasible-region approximation has a 0.223% relative area gap.Six initial conditions near the approximation boundary are used, and the reported trajectories include one representative tube sequence.Source: §5 · p. 8, Figure 1 · p. 8
- The optimal MPC cost decreases along the reported trajectories up to numerical tolerance, consistent with the closed-loop convergence result.Source: §5 · p. 8
5.1 Computational Aspects
- The 95th-percentile solution time rises from 29.3 ms at 6 vertices to 226.4 ms at 20; 10- or 12-vertex templates offer the example's most favorable trade-off.Increasing vertex count enlarges the feasible region with diminishing geometric returns and superlinear computation-time growth.Source: §5.1 · p. 9
6 Conclusions
The paper extends configuration-constrained tube MPC to uncertain nonlinear systems admitting a DC decomposition, with robust propagation and terminal conditions supporting recursive feasibility and convergence to a target robust control invariant set.
6 Conclusions
- The controller jointly optimizes a flexible polytopic tube and its vertex inputs in one convex program, with recursive feasibility and convergence to a target robust control invariant set guaranteed by an implicit terminal condition.Its robust propagation certificate combines convex directional bounds with the affine vertex representation of configuration-constrained polytopes.Source: §6 · p. 9
- Figure 2 plots optimal MPC cost in panel (a) and compares median online computation time with feasible-region approximation area in panel (b).Source: Figure 2 · p. 9
- Future work will consider nonlinear reference tracking and more systematic analysis of geometry and vertex-control restrictions for balancing conservatism and online complexity.Source: §6 · p. 9
A Sign-Preserving Template Transformation
An offline transformation adapts a seed template to the system dynamics while preserving its sign pattern, and can be optimized for geometric criteria such as maximizing the resulting robust control invariant set's volume.
A Sign-Preserving Template Transformation
- The offline transformation adapts the seed template to system dynamics while preserving each facet direction’s sign partition for the DC upper bound.The sign pattern is maintained through restrictions on the transformation, with strict inequalities enforced using a small numerical margin.Source: A Sign-Preserving Template… · pp. 9–10
- For invertible T, the transformed set is X = T^-1 Xbar, and each transformed vertex xj satisfies Txj = xbarj.Source: A Sign-Preserving Template… · p. 9
- Any feasible solution of the offline NLP certifies that X is robust control invariant, and its objective can target the set’s desired geometric properties.The numerical example chooses volume maximization for the resulting RCI set.Source: A Sign-Preserving Template… · pp. 9–10
- Volume maximization can use det T, while a facet-distance surrogate is offered to avoid explicit determinant evaluation.Source: A Sign-Preserving Template… · p. 10
- An invertible transformation preserves the polytope’s face lattice, and the transformed family retains a configuration triple.Source: A Sign-Preserving Template… · p. 10