Reach-Avoid Model Predictive Control with Guaranteed Recursive Feasibility via Input Constrained Backstepping

Jianqiang Ding; Nishant Jayesh Bhave; Shankar A. Deka

arxiv: 2604.03407 · v1 · submitted 2026-04-03 · 🧮 math.OC · cs.SY· eess.SY

Reach-Avoid Model Predictive Control with Guaranteed Recursive Feasibility via Input Constrained Backstepping

Jianqiang Ding , Nishant Jayesh Bhave , Shankar A. Deka This is my paper

Pith reviewed 2026-05-13 18:03 UTC · model grok-4.3

classification 🧮 math.OC cs.SYeess.SY

keywords model predictive controlreach-avoidrecursive feasibilitybacksteppingsampled-datainvariant setcontrol-affine systemsinput constraints

0 comments

The pith

Input-constrained backstepping constructs a terminal set that guarantees recursive feasibility in sampled-data reach-avoid MPC for nonlinear systems.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a model predictive control strategy for continuous-time control-affine nonlinear systems that must reach a desired set while avoiding unsafe regions, all while respecting input constraints. It uses a backstepping method to propagate constraints and build a reach-avoid invariant set that serves as the terminal condition for the MPC optimization. The main result shows that this setup ensures the optimization problem remains feasible at every sampling step, allowing the system to be safely driven to the target when the sampling is sufficiently rapid. This provides rigorous guarantees for safety-critical applications where standard MPC approaches may lose feasibility.

Core claim

By propagating both input and output constraints through an input-constrained backstepping process, a reach-avoid invariant set is synthesized that complies with control input limits. This set is employed as the terminal set in a sampled-data MPC framework, which is proven to recursively admit feasible control inputs that steer the continuous system into the target set under fast sampling conditions.

What carries the argument

The input-constrained backstepping procedure that synthesizes a nonempty reach-avoid invariant set respecting input constraints, serving as the terminal set for the MPC problem.

Load-bearing premise

That the backstepping procedure can synthesize a nonempty reach-avoid invariant set respecting the input constraints for the considered control-affine systems, and that the sampling period is small enough relative to the system dynamics.

What would settle it

Finding a control-affine system where the backstepping-based reach-avoid set cannot be constructed as nonempty within input bounds, or observing loss of feasibility in the MPC optimization for a sampling period that is not sufficiently small.

Figures

Figures reproduced from arXiv: 2604.03407 by Jianqiang Ding, Nishant Jayesh Bhave, Shankar A. Deka.

**Figure 2.** Figure 2: (a), (b) & (c) show controller success rates for [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗

read the original abstract

This letter proposes a novel sampled-data model predictive control framework for continuous control-affine nonlinear systems that provides rigorous reach-avoid and recursive feasibility guarantees under physical constraints. By propagating both input and output constraints through backstepping process, we present a constructive approach to synthesize a reach-avoid invariant set that complies with control input limits. Using this reach-avoid set as a terminal set, we prove that the proposed sampled-data MPC framework recursively admits feasible control inputs that safely steer the continuous system into the target set under fast sampling conditions. Numerical results demonstrate the efficacy of the proposed approach.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper constructs a reach-avoid terminal set for sampled-data MPC via input-constrained backstepping on control-affine systems, but leaves the nonemptiness of that set under tight input bounds without explicit conditions.

read the letter

The main contribution is a constructive method that propagates input and state constraints through backstepping to build a terminal set for nonlinear MPC. This set is then used to enforce both reach-avoid safety and recursive feasibility in the sampled-data setting, with the claim that fast enough sampling preserves the continuous-time guarantees. That pipeline is not standard in the existing MPC or backstepping literature, so the combination is new on its face. The abstract and claimed proofs focus on control-affine dynamics, which keeps the backstepping steps manageable and avoids some of the usual obstacles in general nonlinear systems. Numerical examples are mentioned to show the approach works on concrete cases. The soft spot is the missing sufficient conditions for the terminal set to be nonempty once input bounds are imposed. The stress-test note flags this correctly: if the input limits are small relative to the drift terms, backstepping may produce an empty set even when a reach-avoid controller exists in continuous time. Without explicit bounds on sampling period or input size that guarantee nonemptiness, the recursive feasibility result stays conditional on an assumption that is not automatically true. The paper would still be worth refereeing because the construction is explicit and the safety goal is practically relevant for robotics. A serious reviewer could check the backstepping derivations and ask for the missing nonemptiness criteria or counter-examples. I would bring it to a reading group for the technical details on constraint propagation, but I would not cite it until the nonemptiness issue is resolved.

Referee Report

1 major / 2 minor

Summary. The paper proposes a sampled-data MPC framework for continuous control-affine nonlinear systems. It propagates input and output constraints through an input-constrained backstepping procedure to construct a reach-avoid invariant terminal set, then proves that the resulting MPC problem is recursively feasible and that the closed-loop sampled-data system reaches the target set while avoiding obstacles under sufficiently fast sampling.

Significance. If the nonemptiness and feasibility arguments hold, the work supplies a constructive, constraint-respecting terminal-set design for nonlinear reach-avoid MPC that preserves recursive feasibility; this is a useful addition to the literature on safety-critical sampled-data control.

major comments (1)

[backstepping synthesis of reach-avoid set] The backstepping construction of the reach-avoid invariant set (described after the problem formulation) supplies no explicit sufficient conditions guaranteeing nonemptiness when the input bounds are tight relative to the drift and control vector fields. Because recursive feasibility of the MPC problem is asserted by using this set as the terminal constraint, the absence of such conditions leaves the central claim conditional on an unverified assumption.

minor comments (2)

[main theorem on recursive feasibility] The statement of the sampling-rate condition in the recursive-feasibility theorem should be made fully explicit (e.g., an upper bound on the sampling period expressed in terms of the Lipschitz constants or vector-field norms appearing in the backstepping design).
Notation for the propagated input and output constraint sets should be introduced once and used consistently; several passages reuse similar symbols without clear redefinition.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. The single major comment raises a valid point about the need for explicit nonemptiness conditions on the backstepping-derived reach-avoid set. We address this directly below and will revise the manuscript accordingly to strengthen the central claims.

read point-by-point responses

Referee: The backstepping construction of the reach-avoid invariant set (described after the problem formulation) supplies no explicit sufficient conditions guaranteeing nonemptiness when the input bounds are tight relative to the drift and control vector fields. Because recursive feasibility of the MPC problem is asserted by using this set as the terminal constraint, the absence of such conditions leaves the central claim conditional on an unverified assumption.

Authors: We agree that explicit sufficient conditions for nonemptiness would make the recursive feasibility result fully unconditional and more readily verifiable. In the revised manuscript we will insert a new remark (immediately following the backstepping construction) that supplies such conditions. These conditions are expressed in terms of the Lipschitz constants of the drift and control vector fields, the input bound magnitude, and a sufficiently small sampling period; they guarantee that the reachable set computed by the input-constrained backstepping procedure is nonempty. The proof of recursive feasibility will then be stated under these explicit, checkable assumptions. We have already confirmed that the conditions hold for the numerical example in the paper. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation proceeds from dynamics and constraints

full rationale

The paper constructs a reach-avoid invariant set by propagating input and output constraints through a backstepping procedure applied to the given control-affine dynamics. This set is then used as the terminal constraint in the sampled-data MPC formulation. Recursive feasibility is shown to hold under the assumption of sufficiently fast sampling, without any reduction of the terminal set, feasibility claim, or nonemptiness assertion to a fitted parameter, self-referential definition, or load-bearing self-citation. The central steps rely on explicit propagation from the system vector fields and bounds rather than renaming or smuggling prior results. This is the expected self-contained case for a constructive control synthesis paper.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the existence of a backstepping-derived invariant set for any control-affine system satisfying the input constraints and on the sampling period being small enough for the discrete-time MPC to inherit continuous-time safety. No explicit free parameters, invented entities, or additional axioms are stated in the abstract.

axioms (2)

domain assumption The nonlinear system is control-affine and the backstepping procedure can propagate both state and input constraints to produce a nonempty reach-avoid invariant set.
Invoked when the abstract states that the reach-avoid set is synthesized by propagating constraints through backstepping.
domain assumption Sampling is sufficiently fast relative to the closed-loop dynamics so that the discrete MPC inherits the continuous-time reach-avoid property.
Required for the recursive feasibility proof under fast sampling conditions.

pith-pipeline@v0.9.0 · 5403 in / 1450 out tokens · 58655 ms · 2026-05-13T18:03:10.632302+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

By propagating both input and output constraints through backstepping process, we present a constructive approach to synthesize a reach-avoid invariant set that complies with control input limits.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Using this reach-avoid set as a terminal set, we prove that the proposed sampled-data MPC framework recursively admits feasible control inputs...

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

18 extracted references · 18 canonical work pages

[1]

Experimental vali- dation of safe mpc for autonomous driving in uncertain environments,

I. Batkovic, A. Gupta, M. Zanon, and P. Falcone, “Experimental vali- dation of safe mpc for autonomous driving in uncertain environments,” IEEE Transactions on Control Systems Technology, vol. 31, no. 5, pp. 2027–2042, 2023

work page 2027
[2]

Energy management for smart grids with electric vehicles based on hierarchical mpc,

F. Kennel, D. G ¨orges, and S. Liu, “Energy management for smart grids with electric vehicles based on hierarchical mpc,”IEEE Transactions on industrial informatics, vol. 9, no. 3, pp. 1528–1537, 2012

work page 2012
[3]

Adaptive clf-mpc with application to quadrupedal robots,

M. V . Minniti, R. Grandia, F. Farshidian, and M. Hutter, “Adaptive clf-mpc with application to quadrupedal robots,”IEEE Robotics and Automation Letters, vol. 7, no. 1, pp. 565–572, 2021

work page 2021
[4]

Safety-critical model predictive control with discrete-time control barrier function,

J. Zeng, B. Zhang, and K. Sreenath, “Safety-critical model predictive control with discrete-time control barrier function,” in2021 American Control Conference (ACC), pp. 3882–3889, IEEE, 2021

work page 2021
[5]

Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control,

E. C. Kerrigan and J. M. Maciejowski, “Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control,” inProceedings of the 39th IEEE conference on decision and control (Cat. No. 00CH37187), vol. 5, pp. 4951–4956, IEEE, 2000

work page 2000
[6]

On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach,

J. M. Bravo, D. Lim ´on, T. Alamo, and E. F. Camacho, “On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach,”Automatica, vol. 41, no. 9, pp. 1583– 1589, 2005

work page 2005
[7]

Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach,

B. Decardi-Nelson and J. Liu, “Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach,” Computers & Chemical Engineering, vol. 145, p. 107177, 2021

work page 2021
[8]

Computing controlled invariant sets for hybrid systems with applications to model-predictive control,

B. Legat, P. Tabuada, and R. M. Jungers, “Computing controlled invariant sets for hybrid systems with applications to model-predictive control,”IFAC-PapersOnLine, vol. 51, no. 16, pp. 193–198, 2018

work page 2018
[9]

Nonlinear model predictive control based on k-step control invariant sets,

Z. Zhao, A. Girard, and S. Olaru, “Nonlinear model predictive control based on k-step control invariant sets,”European journal of control, vol. 80, p. 101040, 2024

work page 2024
[10]

Discrete-time control barrier functions for guaranteed recursive feasibility in nonlinear mpc: An application to lane merging,

A. Katriniok, E. Shakhesi, and W. Heemels, “Discrete-time control barrier functions for guaranteed recursive feasibility in nonlinear mpc: An application to lane merging,” in2023 62nd IEEE Conference on Decision and Control (CDC), pp. 3776–3783, IEEE, 2023

work page 2023
[11]

Recursive feasibility guarantees in multi-horizon mpc,

V . N. Behrunani, P. Heer, R. S. Smith, and J. Lygeros, “Recursive feasibility guarantees in multi-horizon mpc,” in2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 297–302, IEEE, 2024

work page 2024
[12]

Model predictive control with preview: Recursive feasibility and stability,

X. Fang and W.-H. Chen, “Model predictive control with preview: Recursive feasibility and stability,”IEEE Control Systems Letters, vol. 6, pp. 2647–2652, 2022

work page 2022
[13]

Reach-avoid controllers synthesis for safety critical systems,

B. Xue, “Reach-avoid controllers synthesis for safety critical systems,” IEEE Transactions on Automatic Control, vol. 69, no. 12, pp. 8892– 8899, 2024

work page 2024
[14]

Backstepping reach-avoid con- troller synthesis for multi-input multi-output systems with mixed relative degrees,

J. Ding, D. Yuan, and S. A. Deka, “Backstepping reach-avoid con- troller synthesis for multi-input multi-output systems with mixed relative degrees,”arXiv preprint arXiv:2505.03612, 2025

work page arXiv 2025
[15]

Semi-global asymptotic stability of a class of sampled-data nonlinear systems in output feedback form,

B. Wu and Z. Ding, “Semi-global asymptotic stability of a class of sampled-data nonlinear systems in output feedback form,” in2008 47th IEEE Conference on Decision and Control, pp. 5420–5425, IEEE, 2008

work page 2008
[16]

H. K. Khalil and J. W. Grizzle,Nonlinear systems, vol. 3. Prentice hall Upper Saddle River, NJ, 2002

work page 2002
[17]

Papachristodoulou, J

A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler, and P. A. Parrilo,SOSTOOLS: Sum of squares optimization toolbox for MATLAB.http://arxiv.org/abs/1310.4716, 2013

work page arXiv 2013
[18]

Mosek optimization toolbox for matlab,

M. ApS, “Mosek optimization toolbox for matlab,”User’s Guide and Reference Manual, Version, vol. 4, no. 1, p. 116, 2019. APPENDIX A. Additional details for the proof of Lemma 2 ∥x(t)−x(t k)∥ ≤ Z t tk f(x(s)) + mX j=1 gj(x(s))uc j(tk) ds ≤ Z t tk ℓf ∥x(s)−x(t k)∥+∥ ˙x(tk)∥ + mX j=1 ℓgj Muc j ∥x(s)−x(t k)∥ ds = Z t tk (ℓf + mX j=1 ℓgj Muc j)∥x(s)−x(t k)∥ ds...

work page 2019

[1] [1]

Experimental vali- dation of safe mpc for autonomous driving in uncertain environments,

I. Batkovic, A. Gupta, M. Zanon, and P. Falcone, “Experimental vali- dation of safe mpc for autonomous driving in uncertain environments,” IEEE Transactions on Control Systems Technology, vol. 31, no. 5, pp. 2027–2042, 2023

work page 2027

[2] [2]

Energy management for smart grids with electric vehicles based on hierarchical mpc,

F. Kennel, D. G ¨orges, and S. Liu, “Energy management for smart grids with electric vehicles based on hierarchical mpc,”IEEE Transactions on industrial informatics, vol. 9, no. 3, pp. 1528–1537, 2012

work page 2012

[3] [3]

Adaptive clf-mpc with application to quadrupedal robots,

M. V . Minniti, R. Grandia, F. Farshidian, and M. Hutter, “Adaptive clf-mpc with application to quadrupedal robots,”IEEE Robotics and Automation Letters, vol. 7, no. 1, pp. 565–572, 2021

work page 2021

[4] [4]

Safety-critical model predictive control with discrete-time control barrier function,

J. Zeng, B. Zhang, and K. Sreenath, “Safety-critical model predictive control with discrete-time control barrier function,” in2021 American Control Conference (ACC), pp. 3882–3889, IEEE, 2021

work page 2021

[5] [5]

Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control,

E. C. Kerrigan and J. M. Maciejowski, “Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control,” inProceedings of the 39th IEEE conference on decision and control (Cat. No. 00CH37187), vol. 5, pp. 4951–4956, IEEE, 2000

work page 2000

[6] [6]

On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach,

J. M. Bravo, D. Lim ´on, T. Alamo, and E. F. Camacho, “On the computation of invariant sets for constrained nonlinear systems: An interval arithmetic approach,”Automatica, vol. 41, no. 9, pp. 1583– 1589, 2005

work page 2005

[7] [7]

Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach,

B. Decardi-Nelson and J. Liu, “Computing robust control invariant sets of constrained nonlinear systems: A graph algorithm approach,” Computers & Chemical Engineering, vol. 145, p. 107177, 2021

work page 2021

[8] [8]

Computing controlled invariant sets for hybrid systems with applications to model-predictive control,

B. Legat, P. Tabuada, and R. M. Jungers, “Computing controlled invariant sets for hybrid systems with applications to model-predictive control,”IFAC-PapersOnLine, vol. 51, no. 16, pp. 193–198, 2018

work page 2018

[9] [9]

Nonlinear model predictive control based on k-step control invariant sets,

Z. Zhao, A. Girard, and S. Olaru, “Nonlinear model predictive control based on k-step control invariant sets,”European journal of control, vol. 80, p. 101040, 2024

work page 2024

[10] [10]

Discrete-time control barrier functions for guaranteed recursive feasibility in nonlinear mpc: An application to lane merging,

A. Katriniok, E. Shakhesi, and W. Heemels, “Discrete-time control barrier functions for guaranteed recursive feasibility in nonlinear mpc: An application to lane merging,” in2023 62nd IEEE Conference on Decision and Control (CDC), pp. 3776–3783, IEEE, 2023

work page 2023

[11] [11]

Recursive feasibility guarantees in multi-horizon mpc,

V . N. Behrunani, P. Heer, R. S. Smith, and J. Lygeros, “Recursive feasibility guarantees in multi-horizon mpc,” in2024 IEEE 63rd Conference on Decision and Control (CDC), pp. 297–302, IEEE, 2024

work page 2024

[12] [12]

Model predictive control with preview: Recursive feasibility and stability,

X. Fang and W.-H. Chen, “Model predictive control with preview: Recursive feasibility and stability,”IEEE Control Systems Letters, vol. 6, pp. 2647–2652, 2022

work page 2022

[13] [13]

Reach-avoid controllers synthesis for safety critical systems,

B. Xue, “Reach-avoid controllers synthesis for safety critical systems,” IEEE Transactions on Automatic Control, vol. 69, no. 12, pp. 8892– 8899, 2024

work page 2024

[14] [14]

Backstepping reach-avoid con- troller synthesis for multi-input multi-output systems with mixed relative degrees,

J. Ding, D. Yuan, and S. A. Deka, “Backstepping reach-avoid con- troller synthesis for multi-input multi-output systems with mixed relative degrees,”arXiv preprint arXiv:2505.03612, 2025

work page arXiv 2025

[15] [15]

Semi-global asymptotic stability of a class of sampled-data nonlinear systems in output feedback form,

B. Wu and Z. Ding, “Semi-global asymptotic stability of a class of sampled-data nonlinear systems in output feedback form,” in2008 47th IEEE Conference on Decision and Control, pp. 5420–5425, IEEE, 2008

work page 2008

[16] [16]

H. K. Khalil and J. W. Grizzle,Nonlinear systems, vol. 3. Prentice hall Upper Saddle River, NJ, 2002

work page 2002

[17] [17]

Papachristodoulou, J

A. Papachristodoulou, J. Anderson, G. Valmorbida, S. Prajna, P. Seiler, and P. A. Parrilo,SOSTOOLS: Sum of squares optimization toolbox for MATLAB.http://arxiv.org/abs/1310.4716, 2013

work page arXiv 2013

[18] [18]

Mosek optimization toolbox for matlab,

M. ApS, “Mosek optimization toolbox for matlab,”User’s Guide and Reference Manual, Version, vol. 4, no. 1, p. 116, 2019. APPENDIX A. Additional details for the proof of Lemma 2 ∥x(t)−x(t k)∥ ≤ Z t tk f(x(s)) + mX j=1 gj(x(s))uc j(tk) ds ≤ Z t tk ℓf ∥x(s)−x(t k)∥+∥ ˙x(tk)∥ + mX j=1 ℓgj Muc j ∥x(s)−x(t k)∥ ds = Z t tk (ℓf + mX j=1 ℓgj Muc j)∥x(s)−x(t k)∥ ds...

work page 2019