pith. sign in

arxiv: 2605.18510 · v1 · pith:UKJDPDC2new · submitted 2026-05-18 · 📡 eess.SY · cs.SY· math.OC

On Piecewise Quadratic Terminal Costs for MPC

Sampath Kumar Mulagaleti , Boris Houska , Mario Zanon , Mario E. Villanueva This is my paper

Pith reviewed 2026-05-20 09:17 UTC · model grok-4.3

classification 📡 eess.SY cs.SYmath.OC
keywords Model Predictive ControlTerminal CostTerminal RegionLinear Quadratic RegulatorPolytopic SetsStability Analysis
0
0 comments X

The pith

MPC terminal cost equals infinite-horizon LQR cost near steady state with new polytopic region

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

This paper develops terminal ingredients for linear model predictive control that aim to enlarge the region of attraction and minimize the gap to the infinite-horizon optimal solution. The approach uses a terminal cost that matches the linear-quadratic regulator cost exactly in a neighborhood of the equilibrium point. It also introduces a terminal region computed via configuration-constrained polytopic methods to ensure positive invariance. Case studies demonstrate improved performance over existing techniques.

Core claim

The central claim is that a piecewise quadratic terminal cost, which is identical to the infinite-horizon LQR cost in a nontrivial neighborhood of the steady-state, combined with a novel terminal region from configuration-constrained polytopic computing, provides stabilizing terminal ingredients for linear MPC. This construction increases the region of attraction while reducing suboptimality compared to standard approaches.

What carries the argument

The piecewise quadratic terminal cost that equals the LQR cost near the steady-state, together with the configuration-constrained polytopic terminal region that ensures invariance under the closed-loop MPC dynamics.

If this is right

  • The MPC feedback law ensures asymptotic stability within the terminal region.
  • Recursive feasibility of the optimization problem is guaranteed for states in the region of attraction.
  • The closed-loop performance approaches that of the infinite-horizon LQR controller near the equilibrium.
  • Comparisons show larger feasible sets than traditional quadratic terminal costs.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method could be adapted for systems with constraints by adjusting the polytopic computations.
  • Extending this to time-varying or uncertain systems might further improve robustness in practical applications.
  • Such terminal costs might reduce the need for long prediction horizons in MPC implementations.

Load-bearing premise

The configuration-constrained polytopic terminal region must remain positively invariant under the closed-loop dynamics induced by the piecewise quadratic terminal cost and the MPC feedback law.

What would settle it

A trajectory starting inside the proposed terminal region that leaves the region under the closed-loop MPC dynamics with the piecewise quadratic cost would falsify the invariance and thus the stability guarantee.

Figures

Figures reproduced from arXiv: 2605.18510 by Boris Houska, Mario E. Villanueva, Mario Zanon, Sampath Kumar Mulagaleti.

Figure 1
Figure 1. Figure 1: (Top) State-space of (36), along with closed-loop trajectory with ut = µ5(xt) in black, and infinite horizon solution in magenta, from initial state indicated by the red dot; (Bottom) Lyapunov function V5(xt). control over the computational complexity of the terminal region. Generally, cc-polytopes with larger numbers of facets, edges, and vertices tend to be more flexible, which leads to larger terminal r… view at source ↗
Figure 2
Figure 2. Figure 2: (Right) Comparison of admissible region size and suboptimality with benchmark approaches. The black lines refer to axis on the left, with the value denoting the Hausdorff distance between the maximal control invariant set XMCI and admissible set ON of the MPC scheme with horizon length N. The axis on the right denotes suboptimality of closed-loop performance of the MPC schemes averaged over 3000 samples fr… view at source ↗
Figure 3
Figure 3. Figure 3: (Top) Comparison of admissible region size; (Bottom) Comparison of suboptimality and solution time. high-complexity maximal control invariant sets, where benchmark approaches often become intractable. Effect of β: We briefly discuss the effect of the β ∈ [0, 1), used in the formu￾lation of the terminal set T(β) in (15), and the terminal cost matrix Θ in (18), on the admissible set and suboptimality. As β →… view at source ↗
Figure 4
Figure 4. Figure 4: (Left) Normalized closed-loop state trajectories with [PITH_FULL_IMAGE:figures/full_fig_p017_4.png] view at source ↗
read the original abstract

This paper presents a novel approach to synthesize stabilizing termi- nal ingredients for linear model predictive control (MPC) schemes, with the aim of increasing the region of attraction while reducing suboptimal- ity with respect to the solution of the infinite-horizon optimal control problem. It is based on the construction of a novel terminal region using methods from the field of configuration-constrained polytopic computing, along with a terminal cost that is exactly equal to the infinite-horizon linear-quadratic regulator cost in a nontrivial neighborhood of the steady- state. The practical performance of the controller is illustrated through various case studies, and comparisons with state-of-the-art approaches are presented.

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.

Referee Report

1 major / 2 minor

Summary. The manuscript proposes a construction of stabilizing terminal ingredients for linear MPC. It defines a terminal region X_f via configuration-constrained polytopic methods and a piecewise-quadratic terminal cost V_f that is identical to the infinite-horizon LQR cost inside a nontrivial neighborhood of the origin. The claimed benefits are an enlarged region of attraction and reduced suboptimality relative to standard terminal-cost choices; numerical case studies and comparisons are included.

Significance. If the positive-invariance property of X_f under the closed-loop MPC law holds, the construction would supply a concrete, computationally tractable route to larger feasible sets while preserving recursive feasibility and stability. The explicit matching to the LQR cost near the origin is a clear advantage over generic quadratic terminal costs.

major comments (1)
  1. [§4] §4 (Terminal-set invariance): the central stability argument requires that the configuration-constrained polytopic set X_f remains positively invariant under the MPC feedback u=κ(x) induced by the piecewise-quadratic V_f. The manuscript supplies only the LQR invariance argument inside the neighborhood where V_f coincides with the LQR cost; no separate invariance proof or numerical verification is given for trajectories that start in X_f but leave the LQR neighborhood under the actual MPC law. This step is load-bearing for the claimed recursive feasibility and stability.
minor comments (2)
  1. The precise definition of the piecewise-quadratic function (how the pieces are chosen and how continuity is enforced at the boundary of the LQR neighborhood) should be stated explicitly, preferably with an equation number.
  2. Figure captions and axis labels in the case-study plots should indicate which controller (proposed vs. baseline) corresponds to each curve.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback on our manuscript. We address the single major comment below and will revise the manuscript to strengthen the relevant arguments.

read point-by-point responses

  1. Referee: [§4] §4 (Terminal-set invariance): the central stability argument requires that the configuration-constrained polytopic set X_f remains positively invariant under the MPC feedback u=κ(x) induced by the piecewise-quadratic V_f. The manuscript supplies only the LQR invariance argument inside the neighborhood where V_f coincides with the LQR cost; no separate invariance proof or numerical verification is given for trajectories that start in X_f but leave the LQR neighborhood under the actual MPC law. This step is load-bearing for the claimed recursive feasibility and stability.

    Authors: We agree that an explicit invariance argument for the full terminal set X_f under the closed-loop MPC feedback is necessary for a complete stability proof. The manuscript establishes positive invariance under the LQR controller inside the neighborhood where V_f coincides with the infinite-horizon LQR cost. In the revised manuscript we will add a dedicated subsection to §4 that proves X_f is positively invariant under the MPC-induced feedback κ(x) for the entire set. The proof exploits the configuration-constrained polytopic representation of X_f together with the descent property of the piecewise-quadratic terminal cost to show that the successor state under the optimal MPC input remains inside X_f. We will also augment the numerical case studies with explicit checks confirming invariance for sample trajectories starting in X_f outside the LQR neighborhood. These additions will close the gap without changing the paper’s main contributions. revision: yes

Circularity Check

0 steps flagged

No circularity: terminal ingredients constructed independently via polytopic methods and LQR matching

full rationale

The paper constructs a terminal region via configuration-constrained polytopic computing and defines the terminal cost to equal the infinite-horizon LQR cost inside a nontrivial neighborhood of the steady-state. No load-bearing step reduces a claimed result to a fitted parameter, self-citation chain, or definitional equivalence inside the paper. The positive-invariance requirement for the terminal set under the piecewise-quadratic MPC law is presented as a property to be ensured by the construction rather than derived tautologically from the inputs. The approach is self-contained against external benchmarks (LQR theory and polytopic invariance checks) with no reduction of the central claims to the paper's own fitted quantities or prior self-citations.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are stated or can be inferred.

pith-pipeline@v0.9.0 · 5643 in / 1092 out tokens · 43463 ms · 2026-05-20T09:17:28.821766+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Lean theorems connected to this paper

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

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

161 extracted references · 161 canonical work pages

  1. [1]

    Artstein, Z. (1983). Stabilization with relaxed controls. Nonlinear Analysis: Theory, Methods & Applications, 7(11), 1163--1173

    work page 1983

  2. [2]

    Badalamenti, F., Mulagaleti, S.K., Villanueva, M.E., Houska, B., and Bemporad, A. (2025). Efficient configuration-constrained tube MPC via variables restriction and template selection

    work page 2025

  3. [3]

    Bemporad, A., Morari, M., Dua, V., and Pistikopoulos, E. (2002). The explicit linear quadratic regulator for constrained systems. Automatica, 38(1), 3--20

    work page 2002

  4. [4]

    and Miani, S

    Blanchini, F. and Miani, S. (2015). Set-theoretic methods in control. Systems & Control: Foundations & Applications. Birkh\"auser

    work page 2015

  5. [5]

    and Allg\"ower, F

    Chen, H. and Allg\"ower, F. (1998). A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(10), 1205--1217

    work page 1998

  6. [6]

    and Cannon, M

    Darup, M.S. and Cannon, M. (2015). A missing link between nonlinear MPC schemes with guaranteed stability. In Conference on Decision and Control, 4977--4983

    work page 2015

  7. [7]

    and Hafstein, S

    Giesl, P. and Hafstein, S. (2015). Review on computational methods for lyapunov functions. Discrete and Continuous Dynamical Systems, Series B, 20(8), 2291--2331

    work page 2015

  8. [8]

    and Pannocchia, G

    Grammatico, S. and Pannocchia, G. (2013). Achieving a large domain of attraction with short-horizon linear mpc via polyhedral lyapunov functions. In 2013 European Control Conference (ECC), 1059--1064

    work page 2013

  9. [9]

    Gr\"une, L. (2009). Analysis and design of unconstrained nonlinear MPC schemes for finite and infinite dimensional systems. SIAM Journal on Control and Optimization, 48(2), 1206--1228

    work page 2009

  10. [10]

    and Cwikel, M

    Gutman, P. and Cwikel, M. (1986). Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states. IEEE Trans. Autom. Control, 31(4), 373--376

    work page 1986

  11. [11]

    Houska, B., M\"uller, M., and Villanueva, M. (2025). Polyhedral control design: Theory and methods. Annual Reviews in Control, 60, 100992

    work page 2025

  12. [12]

    and Taghavian, H

    Johansson, M. and Taghavian, H. (2024). Stable mpc with maximal terminal sets and quadratic terminal costs

    work page 2024

  13. [13]

    Kalman, R.E. et al. (1960). Contributions to the theory of optimal control. Bol. soc. mat. mexicana, 5(2), 102--119

    work page 1960

  14. [14]

    and Engell, S

    Klatt, K.U. and Engell, S. (1998). Gain-scheduling trajectory control of a continuous stirred tank reactor. Computers & Chemical Engineering, 22(4), 491--502

    work page 1998

  15. [15]

    Mulagaleti, S.K., Mejari, M., and Bemporad, A. (2025). Parameter-dependent robust control invariant sets for lpv systems with bounded parameter-variation rate. IEEE Transactions on Automatic Control, 70(2), 1259--1266

    work page 2025

  16. [16]

    and Badgwell, T.A

    Qin, S. and Badgwell, T.A. (2003). A survey of industrial model predictive control technology. Control Engineering Practice, 11(7), 733--764

    work page 2003

  17. [17]

    and Lazar, M

    Raković, S.V. and Lazar, M. (2012). Minkowski terminal cost functions for mpc. Automatica, 48(10), 2721--2725

    work page 2012

  18. [18]

    and Mayne, D

    Rawlings, J. and Mayne, D. (2009). Model Predictive Control: Theory and Design. Madison, WI: Nob Hill Publishing

    work page 2009

  19. [19]

    Rawlings, J., Mayne, D., and Diehl, M. (2017). Model Predictive Control: Theory, Computation, and Design. Nob Hill Publishing

    work page 2017

  20. [20]

    Rockafellar, R. (1970). Convex Analysis. Princeton University Press

    work page 1970

  21. [21]

    Seborg, D., Mellichamp, D., Edgar, T., and Doyle, F. (2010). Process Dynamics and Control. John Wiley & Sons

    work page 2010

  22. [22]

    and Albadawi, H

    Shebrawi, K. and Albadawi, H. (2013). Trace inequalities for matrices. Bull. Aust. Math. Soc., 87, 139--148

    work page 2013

  23. [23]

    and Jaimoukha, I.M

    Tahir, F. and Jaimoukha, I.M. (2013). Robust feedback model predictive control of constrained uncertain systems. Journal of Process Control, 23(2), 189--200. IFAC World Congress Special Issue

    work page 2013

  24. [24]

    Villanueva, M., M \"u ller, M., and Houska, B. (2024). Configuration-constrained T ube MPC . Automatica, 163:111543

    work page 2024

  25. [25]

    and Bemporad, A

    Zanon, M. and Bemporad, A. (2022). C onstrained C ontrol and O bserver D esign by I nverse O ptimality. IEEE Transactions on Automatic Control, 67(10), 5432--5439

    work page 2022

  26. [26]

    Ziegler, G. (1995). Lectures on Polytopes. Springer

    work page 1995

  27. [27]

    Zubov, V. (1965). Methods of A.M. L yapunov and their application. Mathematics of Computation, 19, 349

    work page 1965

  28. [28]

    and Amrit, R

    Angeli, D. and Amrit, R. and Rawlings, J.B. , Title =

    work page

  29. [29]

    and Amrit, R

    Angeli, D. and Amrit, R. and Rawlings, J.B. , Title =. IEEE Trans. Autom. Control , volume =

    work page

  30. [30]

    1983 , author =

    Stabilization with relaxed controls , journal =. 1983 , author =

    work page 1983

  31. [31]

    and Morari, M

    Bemporad, A. and Morari, M. and Dua, V. and Pistikopoulos, E. , title =. Automatica , volume =

    work page

  32. [32]

    Automatica , volume =

    On the minimax reachability of target sets and target tubes , author =. Automatica , volume =

    work page

  33. [33]

    2012 , Address =

    Dynamic Programming and Optimal Control , Author =. 2012 , Address =

    work page 2012

  34. [34]

    Topological Spaces , author =

    work page

  35. [35]

    , title =

    Bitsoris, G. , title =. Systems and Control Letter , volume =

    work page

  36. [36]

    , TITLE =

    Blanchini, F. , TITLE =. Automatica , VOLUME =. 1999 , NUMBER =

    work page 1999

  37. [37]

    and Miani, S

    Blanchini, F. and Miani, S. , TITLE =. 2015 , PAGES =

    work page 2015

  38. [38]

    and Vandenberghe, L

    Boyd, S. and Vandenberghe, L. , title =

    work page

  39. [39]

    , title =

    Borrelli, F. , title =

    work page

  40. [40]

    An introduction to convex polytopes , publisher =

    Br. An introduction to convex polytopes , publisher =

    work page

  41. [41]

    and Allg\"ower, F

    Chen, H. and Allg\"ower, F. , title =. Automatica , volume =

    work page

  42. [42]

    , title =

    Chen, L. , title =. Journal of Approximation Theory , volume =

    work page

  43. [43]

    and Rossiter, J.A

    Chisci, L. and Rossiter, J.A. and Zappa, G. , title =. Automatica , volume =

    work page

  44. [44]

    and Fomin, S.V

    Cornfeld, I.P. and Fomin, S.V. and Sinai, Y.G. and Sossinski, A.B. , title =

    work page

  45. [45]

    , title =

    Federer, H. , title =. Transactions of the American Mathematical Society , volume =

    work page

  46. [46]

    Goulart, P. J. and Kerrigan, E. C. and Maciejowski, J. M. , TITLE =. Automatica , FJOURNAL =. 2006 , NUMBER =

    work page 2006

  47. [47]

    and Cwikel, M

    Gutman, P.O. and Cwikel, M. , title =. IEEE Trans. Autom. Control , volume =

    work page

  48. [48]

    , title =

    Gr\"unbaum, B. , title =

    work page

  49. [49]

    , Journal =

    Gr\"une, L. , Journal =. Analysis and design of unconstrained nonlinear. 2009 , Pages =

    work page 2009

  50. [50]

    and Kvasnica, M

    Herceg, M. and Kvasnica, M. and Jones, C. and Morari, M. , title =. Proceedings of European Control Conference (ECC) , pages =

    work page

  51. [51]

    and Ferreau, H.J

    Houska, B. and Ferreau, H.J. and Diehl, M. , title =. Automatica , year =

    work page

  52. [52]

    2025 , author =

    Polyhedral control design: Theory and methods , journal =. 2025 , author =

    work page 2025

  53. [53]

    Houska, B. and M. On Stabilizing Terminal Costs and Regions for Configuration-Constrained Tube. IEEE Control Systems Letters , year=

    work page

  54. [54]

    and Grieder, P

    Jones, C.N. and Grieder, P. and Rakovi\'c, S. , title =. Automatica , volume =

    work page

  55. [55]

    , title =

    Kerrigan, E.C. , title =

    work page

  56. [56]

    ohler, J. and Andina, E. and Soloperto, R. and M\

    K\"ohler, J. and Andina, E. and Soloperto, R. and M\"uller, M.A. and Allg\"ower, F. , booktitle=. Linear robust adaptive model predictive control: Computational complexity and conservatism , pages=

    work page

  57. [57]

    and Balakrishnan, V

    Kothare, V. and Balakrishnan, V. and Morari, M. , title =. Automatica , volume =

    work page

  58. [58]

    and Cannon, M

    Kouvaritakis, B. and Cannon, M. , title =

    work page

  59. [59]

    Kurzhanski, A. B. and V. Ellipsoidal calculus for estimation and control , SERIES =. 1997 , PAGES =

    work page 1997

  60. [60]

    and Chryssochoos, I

    Langson, W. and Chryssochoos, I. and Rakovi. Robust model predictive control using tubes , JOURNAL =. 2004 , NUMBER =

    work page 2004

  61. [61]

    and Kearfott, R.B

    Moore, R.E. and Kearfott, R.B. and Cloud, M.J. , title =. 2009 , publisher =

    work page 2009

  62. [62]

    , title =

    McMullen, P. , title =. Mathematika , volume =

    work page

  63. [63]

    2003 , author =

    A survey of industrial model predictive control technology , journal =. 2003 , author =

    work page 2003

  64. [64]

    , title =

    Shebrawi, K and Albadawi, H. , title =. Bull. Aust. Math. Soc. , volume =

    work page

  65. [65]

    , title =

    Schneider, R. , title =. Mathematische Annalen , volume =

    work page

  66. [66]

    and Kouvaritakis, B

    Rakovi\'c, S.V. and Kouvaritakis, B. and Cannon, M. and Panos, C. and Findeisen, R. , title =. Transactions on Automatic Control , volume =

    work page

  67. [67]

    and Kouvaritakis, B

    Rakovi\'c, S.V. and Kouvaritakis, B. and Cannon, M. , title =. Systems & Control Letters , volume =

    work page

  68. [68]

    Model Predictive Control: Theory and Design , Author =

    work page

  69. [69]

    Automatica , volume =

    Robust. Automatica , volume =. 2017 , author =

    work page 2017

  70. [70]

    , Title =

    Willems, J.C. , Title =. Archive for Rational Mechanics and Analysis , volume =

    work page

  71. [71]

    , Title =

    Willems, J.C. , Title =. IEEE Trans. Autom. Control , Volume =

    work page

  72. [72]

    2008 , publisher=

    A course on Borel sets , author=. 2008 , publisher=

    work page 2008

  73. [73]

    , title =

    Stanley, R.P. , title =. Advances in Mathematics , volume =

    work page

  74. [74]

    2009 , publisher=

    Set-valued analysis , author=. 2009 , publisher=

    work page 2009

  75. [75]

    Systems & Control Letters , volume=

    Cost-to-travel functions: a new perspective on optimal and model predictive control , author=. Systems & Control Letters , volume=. 2017 , publisher=

    work page 2017

  76. [76]

    and Rawlings, J.B

    Pannocchia, G. and Rawlings, J.B. and Wright, S.J. , journal=. Conditions under which suboptimal nonlinear. 2011 , publisher=

    work page 2011

  77. [77]

    Automatica , volume=

    Inherent robustness properties of quasi-infinite horizon nonlinear model predictive control , author=. Automatica , volume=. 2014 , publisher=

    work page 2014

  78. [78]

    and Villanueva, M.E

    Houska, B. and Villanueva, M.E. , title =. Handbook of Model Predictive Control , publisher =. 2019 , pages =

    work page 2019

  79. [79]

    Angeli, D. and M. Economic Model Predictive Control: Some Design Tools and Analysis Techniques , editor =. Handbook of Model Predictive Control , publisher =. 2019 , pages =

    work page 2019

  80. [80]

    IEEE Trans

    On necessity and robustness of dissipativity in economic model predictive control , author=. IEEE Trans. Autom. Control , volume=. 2015 , publisher=

    work page 2015

Showing first 80 references.