Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive Control

Christophe Gallesco; Diego F. de Bernardini; Luz A. Alvarez

REVIEW 2 major objections 4 minor 30 references

Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive Control

T0 review · 2 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read The paper proves that for the extended infinite-horizon MPC and its zone-control variant, the optimal cost converges to zero and the closed loop is uniformly stable for any input horizon and any (possibly singular) static gain matrix.

desk verdict A solid proof extension of the Odloak MPC stability results; the main claim is likely right, but the paper needs a real proof of the load-bearing φ>0 claim before it is complete. read the letter →

arxiv 2411.14370 v1 pith:YPWMSXGE submitted 2024-11-21 math.OC

Luz A. Alvarez , Diego F. de Bernardini , Christophe Gallesco This is my paper · ORCID

classification math.OC MSC 93C9593D20

keywords modelpredictivecontrolinfinitehorizonnominalstabilityconvergencezoneOPOMnon-regulargainmatrix

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper supplies complete proofs of two facts that earlier analyses of these infinite-horizon MPC algorithms had only argued in restricted settings. For the extended nominally stable MPC with the output prediction-oriented (OPOM) model, under stability of the open-loop system and reachability of the set-point, the optimal cost $V_k^*$ can be driven to zero as $k\to\infty$ by choosing the slack-weighting matrix $S$ appropriately, and the tracking error is uniformly stable. The same conclusions hold for the zone-control variant when the target weight $S_u$ is chosen large enough. These results hold for any input horizon $m$ and for any static gain matrix $D_0$, which need not be square or full rank. This matters because industrial MPC implementations commonly have more outputs than inputs or linearly dependent gains, and earlier proofs assumed a regular $D_0$ and $m=1$.

What carries the argument

The load-bearing object is the finite-horizon equivalent of the infinite-horizon cost together with a specially built slack penalty matrix $S$. Because the open-loop system is stable, the infinite sum in $V_k$ collapses to the finite sum (4) with terminal weight $\bar Q$ solving the Lyapunov relation $\bar Q - F^T\bar QF = F^T\Psi^T Q\Psi F$; the terminal constraint $x_s(m-1|k) - \delta_k - r = 0$ keeps the tail bounded. The matrix $\hat S = K_1^T (L^{-1})^T L^{-1} K_1 + K_2^T K_2$, built from an LQ factorization of $D_0^\perp$, is the device that converts output-side slack costs into input-side distances. The other mechanism is the monotonicity lemma (Lemma 2.5), which shows $V_{k+1}^* \le \tilde V_{k+1} \le V_k^*$ by shifting the optimal move sequence forward, and from it the paper derives $\Delta u^*(0|k)\to 0$ and $x_d^*(0|k)\to 0$, the decay facts needed for the contradiction.

What would settle it

For a stable plant with singular $D_0$, choose a rectangle $U$ with a side parallel to the affine set $U_r = u_r + \ker D_0$, and compute $\phi := \inf_{x\in U\setminus\ker D_0}|\cos\theta_x|$. If $\phi = 0$ and the closed-loop simulation from a nonzero steady state shows $\limsup_{k\to\infty}\|u^*(0|k)-P_r u^*(0|k)\| > 0$ (equivalently, $V_k^*$ does not tend to 0) for some horizon $m\ge 2$, the central claim would be false; at minimum such an example would pinpoint the missing hypothesis in Proposition 2.8.

Watch

Extended reading notes

Core claim

The central discovery is that a single geometric-algebraic construction removes the two standing restrictions on these controllers. Theorem 2.3 states that under the stability and set-point assumptions one can pick $S = \beta\hat S$ so that $\lim_{k\to\infty} V_k^* = 0$, and Theorem 2.4 derives uniform stability of the output error. The construction of $\hat S$ via the LQ factorization of the restriction $D_0^\perp$ makes $\|D_0 v\|_{\hat S} = \|v\|$ for every input direction $v\in(\ker D_0)^\perp$, so that distances in output space become distances in input space. This reduction permits a contradiction argument in Proposition 2.8: if the implemented input sequence stays away from the reference affine set $U_r = u_r + \ker D_0$, a feasible one-step move toward $U_r$ produces a cost strictly lower than the optimum. The zone-control analog, Theorems 3.3 and 3.4, extends the same mechanism to the formulation with output zones and input targets, with $S_u > H + I_{n_u}$ replacing the condition on $S$.

Load-bearing premise

The proof's convergence argument rests on an unproved geometric assertion that the angle between the direction from any feasible input $x$ to its orthogonal projection onto the target affine set and the direction from $x$ to the closest feasible target point is uniformly bounded away from 90 degrees over the rectangle $U$; if this angle can approach 90 degrees, the bound on the corrective input move collapses.

Editorial extensions

If this is right

Recursive feasibility and a non-increasing optimal cost hold for any input horizon $m$, so a practitioner can choose longer horizons without losing the Lyapunov-based stability guarantee.
The explicit choice $S = \beta\hat S$ (with $\beta$ a computable constant) and the zone-control choice $S_u > H + I_{n_u}$ provide concrete tuning rules for the slack weights.
Systems with non-square or singular static gain can be handled directly, covering plants with more outputs than inputs or with redundant actuator directions.
The proof templates in Sections 2 and 3 can be adapted to the derived MPC variants mentioned in the introduction (integrating systems, dead time, unstable systems, two-layer RTO-MPC), exactly as the authors claim.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Beyond the paper's claims, the same projection-and-LQ construction may yield closed-form slack weights for robust or invariant-set MPC variants, since the only model-dependent objects are $D_0$, $\Psi$, $F$, and the Lyapunov weight.
A testable extension not stated in the paper is to allow a set-point sequence $r_k$ that converges geometrically; the proof's estimates suggest the tracking error would inherit the same geometric decay rate.
The unresolved geometric assertion $\phi>0$ is likely the first point to break in practice: for bounded rectangles it is true whenever the target affine subspace is not parallel to a face, but for unbounded $U$ the claimed uniform bound may fail, and the convergence conclusion would need separate hypotheses.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The paper extends Odloak 2004 and Gonzalez-Odloak 2009 convergence/stability proofs from regular D0 and m=1 to any D0 and any input horizon. The generalization matters because non-square gain matrices are the norm in real refineries. The main proof skeleton is sound: Lemma 2.5's cost decrease is real, Lemma 2.6's use of the decay of F is legitimate, and the LQ-factorization construction of S is a genuinely new device that makes the slack-weight choice explicit.

The soft spot is Proposition 2.8's claim that φ := inf |cos θ_x| > 0 over the rectangle U. That infimum is exactly the ratio of the normal distance to Ur to the distance to U∩Ur. It is not a routine consequence of 'rectangle.' For bounded U it should follow from a compactness plus face-by-face transversality argument; for unbounded U you need to be careful about directions at infinity. The claim is probably true, but the paper simply asserts it and the constant C3 in the appendix carries a φ^{-2} factor, so the convergence theorem collapses without it. A referee should ask for a lemma: either assume U bounded and prove the infimum is positive, or state and prove the general polyhedral version. This is fixable, not fatal.

Minor issues: several 'after some elementary computation' steps are actually lengthy substitutions, and a few inequalities in Theorem 2.3 are written as if they follow immediately from Lemma 2.6/2.7 when they need a subsequence argument. These are cosmetic once you fill them in. No code or numerics, but this is a proof paper and doesn't need them. Citations trace the Odloak lineage properly; prior work is credited and the new theorems are clearly marked as not in the earlier papers.

Bottom line: the central claim is plausible and valuable, the proof framework is transparent, and the one load-bearing gap is a missing geometric lemma rather than a hidden circularity. Send it to a serious referee, and tell them to make the authors prove φ > 0 properly. I'd bring it to a reading group, and I'd cite it if I worked on MPC stability theory.

Referee Report

2 major / 4 minor

Summary. The paper analyzes the extended infinite-horizon MPC of Odloak (2004) and the zone-control MPC of González and Odloak (2009). It claims convergence of the optimal cost to zero and uniform stability of the tracking error under a non-regular static gain D0 and for any input horizon m. The proofs use a Lyapunov-type decrease argument, a constructed slack weight S (or Su), and a contradiction argument based on an improved feasible strategy.

Significance. If correct, the results provide the first rigorous convergence and stability proof for these industrially used MPC formulations without requiring D0 to be square or regular, and without restricting the input horizon to m=1. The paper gives explicit constructive conditions on S and Su, and many of the algebraic steps, including the decrease lemmas and the use of stability of F, are checkable and sound. However, the central convergence theorem rests on an unproved geometric claim, and the feasibility arguments for the increment constraints are incomplete.

major comments (2)

[§2.2, Proposition 2.8] The proof asserts without proof that for a rectangle U, ϕ := inf_{x∈U\kerD0} |cos θ_x| > 0, where θ_x is the angle between Pr x − x and Π_r x − x. This is load-bearing: it is used to bound ‖Δũ(0|k)‖ ≤ αϕ^{-1}(c+ε), and the Appendix defines C3 with a ϕ^{-2} factor. The claim is not a routine consequence of U being a rectangle, because kerD0 is an arbitrary subspace, not necessarily coordinate-aligned. Moreover, U is not stated to be bounded, so the infimum can be approached at infinity. I am not claiming the assertion is false; but as written it is an unproved global transversality/error-bound property. Please supply a proof (for example via recession cones and the positive angle between kerD0 and the relevant faces of U) or add boundedness of U with a compactness-based argument. Without this, the contradiction in Proposition 2.8 and hence Theorem 2.3 is incomplete.
[§2.2, Proposition 2.8 and §3.2, Theorem 3.3] Feasibility of the constructed increment needs more care. In Proposition 2.8 the strategy sets Δũ(0|k)=α(Π_r u^*(0|k−1)−u^*(0|k−1)) and the text says 'Since U is convex, we can choose α small enough such that this strategy is feasible.' Feasibility also requires Δũ(0|k)∈ΔU, and ΔU is only assumed to be a rectangle containing the origin. If 0 is not in the interior of ΔU, no positive scaling of that vector may be admissible. Since α is later fixed as 3ε/(c+ε), the choice of ε must be coordinated with the size of ΔU. The same issue appears in Theorem 3.3, where (12) uses Δu^*(m−1|k)∈int ΔU and the perturbed increment Δu^*(m−1|k)−(1−α)δ^*_{u,k} is required to lie in ΔU. Please state explicitly that ΔU (and, where needed, U) have nonempty interior containing the origin, or otherwise justify the existence of the required α.

minor comments (4)

[§2] The convention for x_s(0|k) should be stated explicitly. The equations in Lemmas 2.5 and 2.7 suggest that x_s(0|k) is the state after the first move Δu^*(0|k) has been applied, rather than the measured state before the optimization. With the standard MPC convention, identities such as e_s^*(0|k)−δ_k^* = −D_0∑_{j=1}^{m−1}Δu^*(j|k) would appear inconsistent with (3). Stating the convention would remove ambiguity.
[Acknowledgements] There are typos in the acknowledgements: 'ﬁnantial' should be 'financial'.
[Appendix] The notation Γ_Z^2 ∨ (2Γ_{\bar Q}Γ_{Z−R}) should define the symbol ∨, or use max, for readers.
[Abstract] The phrase 'op erate' in the abstract appears to be a broken word from typesetting; please correct in the final version.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the proofs are self-contained derivations from the model and cost; the unproved geometric assertion in Proposition 2.8 is a rigor gap, not a circular step.

full rationale

The derivation chain is self-contained. Lemmas 2.5, 2.6, 2.7, 3.5, 3.6, 3.7 and Corollary 3.8 are proved directly from the OPOM state equations (1)-(2), the cost definitions, and the optimization constraints; the results of Odloak (2004) and Gonzalez-Odloak (2009) are re-proved 'for the sake of completeness' (Lemma 2.5, Lemma 3.5), so they are not used as black boxes. The weight matrices are not fitted to data or to the conclusion: Proposition 2.8 constructs S = beta*S-hat with an explicit sufficient condition beta > 6*C3, and Theorem 3.3 requires Su > H + I_nu, both being design inequalities expressed in model parameters; no parameter is estimated from a subset of the convergence/stability data and then 'predicted'. I flag one missing support that is not circular: in the proof of Proposition 2.8, the text asserts 'Since U is a rectangle we have that phi := inf_{x in U excluding ker D0} |cos theta_x| > 0' (Section 2.2), and this bound enters the contradiction argument via ||Delta u_tilde(0|k)|| <= alpha*phi^{-1}(c+epsilon) and the Appendix constant C3 = 2*phi^{-2}[Gamma_Z^2 or (2*Gamma_Qbar*Gamma_{Z-R})]. This is a load-bearing geometric transversality claim about U, ker D0, and the projection Pi_r, and it is not proved; it is a correctness/rigor risk, but it is not a reduction of the theorem to its own inputs. Self-citations are contextual and re-derived, so the circularity score is 0.

Assumptions & free parameters 2 free parameters · 5 assumptions · 0 invented entities

The central claims rest on standard MPC assumptions (stable plant, reachable reference, box constraints) plus one nonstandard geometric assertion about projection angles that is not proven. The only free parameters are the slack weight matrices, which the theorems require to be sufficiently large. No new physical entities are introduced.

free parameters (2)

S (slack weight matrix) = S = beta * S_hat with beta > 6C3
The convergence theorem requires the slack penalty to be large enough; the proof constructs S_hat via LQ factorization and gives a lower bound on beta, but no automatic tuning rule.
Su (zone-control input slack weight) = Su > H + I_nu
Theorem 3.3 requires the input slack penalty to exceed a matrix bound H built from model data; this is a design condition, not a fitted value.

assumptions (5)

domain assumption Assumption 2.1 (and 3.1): the open-loop system is stable, spectral radius of F is strictly less than 1.
Used to define finite terminal weight Q_bar and to show x_d dynamics vanish; standard in infinite-horizon MPC for stable systems.
domain assumption Assumption 2.2: the reference r is reachable, r = D0 u_r for some u_r in U. Assumption 3.2: u_des in U and D0 u_des in Y.
Guarantees the target affine subspace intersects the constraint set, so the projection Pi_r exists; without it, offset-free tracking is not expected.
ad hoc to paper Geometric lemma: for a rectangle U, phi := inf_{x in U \ kerD0} |cos theta_x| > 0, where theta_x is the angle between the orthogonal projection direction onto U_r and the constrained projection direction onto U cap U_r.
Stated without proof in Proposition 2.8 and used essentially; it is not a standard textbook lemma and may be false for unbounded rectangles.
domain assumption U, Delta U and Y are fixed rectangles containing the origin.
Standard MPC box constraints; used for convexity and for the angle bound.
standard math The weight matrices Q, R, S (resp. Qy, Qu, R, Sy, Su) are positive definite, so the cost is strictly convex and the optimizer is unique.
Used to identify the optimal solution and for the parallelogram-type inequalities in Theorem 2.4.

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Pith. "Pith review of Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive Control." pith.science (2026). https://pith.science/paper/YPWMSXGE

@misc{pith2026241114370,
  author       = {Pith},
  title        = {Pith review of: Convergence and Stability Analysis of the Extended Infinite Horizon Model Predictive Control},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YPWMSXGE}},
  note         = {Machine review of arXiv:2411.14370}
}

abstract

Model Predictive Control (MPC) is a popular technology to operate industrial systems. It refers to a class of control algorithms that use an explicit model of the system to obtain the control action by minimizing a cost function. At each time step, MPC solves an optimization problem that minimizes the future deviation of the outputs which are calculated from the model. The solution of the optimization problem is a sequence of control inputs, the first input is applied to the system, and the optimization process is repeated at subsequent time steps. In the context of MPC, convergence and stability are fundamental issues. A common approach to obtain MPC stability is by setting the prediction horizon as infinite. For stable open-loop systems, the infinite horizon can be reduced to a finite horizon MPC with a terminal weight computed through the solution of a Lyapunov equation. This paper presents a rigorous analysis of convergence and stability of the extended nominally stable MPC developed by Odloak [Odloak, D. Extended robust model predictive control, AIChE J. 50 (8) (2004) 1824-1836] and the stable MPC with zone control [Gonz\'alez, A.H., Odloak, D. A stable MPC with zone control, J. Proc. Cont. 19 (2009) 110-122]. The mathematical proofs consider that the system is represented by a general gain matrix $D_0$, i.e., not necessarily regular, and they are developed for any input horizon $m$. The proofs are based on elementary geometric and algebraic tools and we believe that they can be adapted to the derived MPC approaches, as well as future studies.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

Alvarez, E.M

L.A. Alvarez, E.M. Francischinelli, B.F. Santoro, D. Od loak, Stable Model Predictive Con- trol for Integrating Systems with Optimizing Targets, Indu strial & Engineering Chemistry Research 48 (2009) 9141-9150

work page 2009

[2] [2]

Alvarez, D

L.A. Alvarez, D. Odloak, Robust integration of real time optimization with linear model predictive control, Computers and Chemical Engineering 34 (2010) 1937-1944

work page 2010

[3] [3]

Alvarez, D

L.A. Alvarez, D. Odloak, Reduction of the QP-MPC cascade structure to a single layer MPC, Journal of Process Control 24 (2014) 1627-1638

work page 2014

[4] [4]

Camacho, C

E.F. Camacho, C. Bordons, Model Predictive Control, 2nd Ed., Springer London (2007)

work page 2007

[5] [5]

Carrapiço, D

O.L. Carrapiço, D. Odloak, A stable model predictive con trol for integrating processes, Computers and Chemical Engineering 29 (2005) 1089-1099

work page 2005

[6] [6]

Carrapiço, M.M

O.L. Carrapiço, M.M. Santos, A.C. Zanin, D. Odloak, Appl ication of the IHMPC to an industrial process system, IF AC Proceedings Volumes 42 (20 09) 851-856

work page

[7] [7]

de Carvalho, L.A

R.F. de Carvalho, L.A. Alvarez, Simultaneous Process De sign and Control of the Williams- Otto Reactor Using Inﬁnite Horizon Model Predictive Contro l, Industrial & Engineering Chemistry Research 59 (2020) 15979-15989

work page 2020

[8] [8]

Costa, L

E.A. Costa, L. Schnitman, D. Odloak, M.A.F. Martins, A On e-Layer Stabilizing Model Predictive Control Strategy of Integrating Systems with Re peated Poles, Journal of Con- trol, Automation and Electrical Systems 33 (2022) 369-381

work page 2022