A Quantitative Framework for Navigating Controller Design Tradeoffs under Computational Constraints

Chris Verhoek; Nikolai Matni

arxiv: 2604.24897 · v2 · submitted 2026-04-27 · 📡 eess.SY · cs.SY

A Quantitative Framework for Navigating Controller Design Tradeoffs under Computational Constraints

Chris Verhoek , Nikolai Matni This is my paper

Pith reviewed 2026-05-15 06:22 UTC · model grok-4.3

classification 📡 eess.SY cs.SY

keywords incremental input-to-state stabilitysmall-gain theoremcontroller approximationcomputational constraintsreceding horizon controlLQRdesign tradeoffssector bound

0 comments

The pith

Approximations in controller design reduce to verifying a sector bound on policy difference from an ideal baseline, with stability via small-gain.

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

The paper develops a framework that treats computational approximations such as model reduction, discretization, horizon truncation, and solver limits as explicit tunable parameters in controller design. It shows that the combined effects of these approximations on closed-loop performance can be bounded using incremental input-to-state stability by checking a single design-dependent sector condition on the difference between the actual policy and an idealized one. Stability of the resulting system is then certified through a small-gain argument. These relations are used to pose a Design Meta-Problem that minimizes performance loss while respecting stability and real-time compute constraints. The method is applied to a receding-horizon LQR example to demonstrate quantitative navigation of tradeoffs among sampling rate, model order, horizon length, and solver iterations.

Core claim

By leveraging incremental input-to-state stability, bounding the aggregate effects of these approximations reduces to verifying a design-dependent sector bound on the difference between the deployed policy and an idealized baseline, with stability enforced via a small-gain condition. This allows the approximations to be treated as design variables inside an optimization problem that minimizes the performance gap subject to stability, real-time compute, and timing constraints.

What carries the argument

Incremental input-to-state stability together with a design-dependent sector bound on policy mismatch, certified by a small-gain theorem.

If this is right

The performance gap between approximate and ideal controllers becomes a quantity that can be minimized under explicit stability and compute constraints.
Sampling rate, model order, horizon length, and solver iterations turn into jointly optimizable design variables.
Stability of the deployed controller follows directly once the sector bound and small-gain condition are verified.
The framework supplies a concrete Design Meta-Problem whose solution yields near-optimal controllers for a given computational budget.

Where Pith is reading between the lines

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

The same bounding technique could be applied to other control architectures if analogous incremental stability margins can be established for those policies.
Controller synthesis software could incorporate automatic selection of approximation levels to satisfy real-time requirements rather than requiring manual tuning.
Extension to plants with significant nonlinearity or uncertainty would require generalizing the sector-bound verification step while preserving the small-gain structure.

Load-bearing premise

The combined impact of model reduction, discretization, horizon truncation, and solver inaccuracy can be captured by one design-dependent sector bound on the difference between the deployed policy and an ideal baseline.

What would settle it

A closed-loop system in which the sector bound holds yet instability occurs, or in which the sector bound is violated yet the system remains stable and meets performance specifications.

Figures

Figures reproduced from arXiv: 2604.24897 by Chris Verhoek, Nikolai Matni.

**Figure 1.** Figure 1: Computed values of the design-dependent sector bounds L ⋆ and Lˆ for 10 randomly selected τ s (∗) for the reduced order models M1, M2, M3, and the fitted function (21) of the sector bound ( ). The right-hand side of the small-gain constraint ( ) in (22b) indicates the feasible region for τ . The vertical blue line ( ) depicts the τ that minimizes ∥Alqrxt − (Axt + BKT ⋆,τx[t] )∥ for all xt . 14 view at source ↗

**Figure 2.** Figure 2: Computed values of the design-dependent sector bound over a grid of τ × ρ for the optimal and deployed systems for system M2. The region for which the small-gain condition (22b) is satisfied is indicated with green. the difference between the deployed and baseline closed-loop systems, rather than the difference between deployed and baseline policies. While appealing, this perspective heavily exploits the l… view at source ↗

**Figure 3.** Figure 3: Simulation of the baseline ( ) and deployed closed-loop systems, where the selected sampling time τ corresponds to either the locally optimal τ ( ), obtained using nonlinear optimization, or the τ that minimizes (22a) ( ). The left-hand plots show ∥xt∥2, with xt ∈ R 97. The right-hand plots shows the approximation of the closed-loop cumulative cost, i.e., R t 0 c(xs, us)ds. under the assumption that G, G … view at source ↗

**Figure 4.** Figure 4: Box plot showing the distribution of the suboptimality gap of the deployed policies corresponding to M1, M2, M3, based on a thousand 10-second simulations with initial conditions on the unit ball, i.e., ∥x0∥ = 1. The deployed policies are designed with a sampling time τ ⋆ , obtained using nonlinear optimization or the DMP (22). Outliers in the distribution are indicated by ◦. 31 view at source ↗

**Figure 5.** Figure 5: Plots demonstrating the effect of choosing linear dependence for Lτ on τ and restricting the exponential rate to αi on the functional form (21) ( ). For the reduced-order models M1, M2, M3, we fitted here the obtained functional form again, now with quadratic dependence for Lτ on τ ( ), and αi as a fittable parameter ( ). We used the same 10 randomly selected points (∗) from the computed “true” suboptimal… view at source ↗

**Figure 6.** Figure 6: Evolution of L ⋆ and Lˆ when the computational resources grow. We showed here the computed values of the design-dependent sector bound over a grid of τ × ρ for the optimal and deployed systems for system M2 and a given τg. As in view at source ↗

read the original abstract

Computational constraints permeate the controller design process, and yet are rarely treated as explicit design constraints. Towards addressing this gap, we propose a quantitative framework that captures the effects of common design approximations, such as model order reduction, temporal discretization, horizon truncation, and solver accuracy, on both controller performance and computational requirements. Our framework highlights that these approximations are tunable parameters within an overall controller design process. By leveraging incremental input-to-state stability, we show that bounding the aggregate effects of these approximations reduces to verifying a design-dependent sector bound on the difference between the deployed policy and an idealized baseline, with stability enforced via a small-gain condition. We operationalize these insights via a Design Meta-Problem in which the performance gap is minimized subject to stability, real-time compute, and timing constraints. Finally, we instantiate the framework on a receding horizon LQR case study, and demonstrate a principled near-optimal navigation of tradeoffs among sampling rate, model order, horizon length, and solver iterations.

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 frames computational approximations as tunable design parameters via incremental ISS and a meta-optimization, which is a practical step forward, but the sector-bound argument needs an explicit projection construction when model reduction changes state dimension.

read the letter

The core contribution is a quantitative setup that treats model reduction, discretization, horizon truncation, and solver accuracy as explicit knobs inside the design process rather than after-the-fact patches. It reduces their combined effect to a design-dependent sector bound on the difference between the deployed policy and an ideal one, then uses a small-gain condition for stability and solves a meta-problem that trades performance against real-time compute and timing limits. The LQR receding-horizon case study shows how to navigate sampling rate, model order, horizon length, and iteration count in one optimization, which is the part that feels immediately usable for embedded control work.

Referee Report

1 major / 1 minor

Summary. The paper proposes a quantitative framework for incorporating computational constraints into controller design. It models the aggregate effects of approximations (model-order reduction, discretization, horizon truncation, solver accuracy) on closed-loop performance by leveraging incremental input-to-state stability and small-gain theorems. The central reduction shows that stability is ensured once a design-dependent sector bound on the difference between the deployed policy and an idealized baseline is verified. The framework is then cast as a Design Meta-Problem minimizing performance gap subject to stability, real-time compute, and timing constraints, and is instantiated on a receding-horizon LQR case study that illustrates trade-offs among sampling rate, model order, horizon length, and solver iterations.

Significance. If the framework is rigorously established, it supplies a principled, quantitative method for navigating performance-computation trade-offs that are ubiquitous in real-time control yet usually treated heuristically. The use of standard incremental-ISS and small-gain machinery to obtain an explicit sector-bound condition is a clear strength, as is the operationalization into an optimizable meta-problem and the concrete LQR demonstration. The result would be of interest to the systems-and-control community working on approximate or resource-aware controller synthesis.

major comments (1)

[Framework derivation] Framework section (derivation of aggregate bound): The reduction of all approximation effects to a single sector bound on ||π_deployed(x) − π_ideal(x)|| assumes that the policy difference is well-defined on a common state space. Model-order reduction changes the state dimension, so an explicit lifting or projection operator must be introduced and shown to be sector-bounded (or Lipschitz with a constant independent of the design parameters) for the subsequent small-gain argument to apply to the composite interconnection. The LQR case study uses a linear system where such an operator can be chosen naturally, but the general statement does not exhibit the required construction or verify that the composite sector constant remains finite for arbitrary reduction maps. This point is load-bearing for the central claim.

minor comments (1)

[Abstract / Introduction] The abstract and introduction would benefit from a brief statement of the precise assumptions under which the sector bound is design-dependent but still verifiable in practice.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and for identifying this important technical point in the framework derivation. We address the comment below and will revise the manuscript accordingly.

read point-by-point responses

Referee: [Framework derivation] Framework section (derivation of aggregate bound): The reduction of all approximation effects to a single sector bound on ||π_deployed(x) − π_ideal(x)|| assumes that the policy difference is well-defined on a common state space. Model-order reduction changes the state dimension, so an explicit lifting or projection operator must be introduced and shown to be sector-bounded (or Lipschitz with a constant independent of the design parameters) for the subsequent small-gain argument to apply to the composite interconnection. The LQR case study uses a linear system where such an operator can be chosen naturally, but the general statement does not exhibit the required construction or verify that the composite sector constant remains finite for arbitrary reduction maps. This point is load-bearing for the central claim.

Authors: We agree that the general statement requires an explicit construction to handle state-dimension mismatch under model-order reduction. In the revised manuscript we will augment the Framework section with the following: we introduce a design-independent projection operator P that embeds the reduced-order state into the full-order space (via modal truncation or least-squares embedding, standard in MOR) and prove that, whenever the reduction error is bounded by a constant independent of the design parameters (as assumed throughout the paper), P is Lipschitz with constant 1 + ε where ε is the reduction tolerance. This ensures the composite sector bound on ||π_deployed − π_ideal|| remains finite, so the small-gain argument continues to apply. We will also spell out the explicit form of P for the LQR case study (state truncation) and verify that the resulting sector constant is finite and independent of the design variables. This addition strengthens rather than alters the central claim. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation applies standard incremental ISS and small-gain theorems to reduce aggregate effects to a verifiable sector bound

full rationale

The paper states that bounding approximation effects 'reduces to verifying a design-dependent sector bound on the difference between the deployed policy and an idealized baseline, with stability enforced via a small-gain condition.' This reduction follows from applying external, established results on incremental input-to-state stability rather than any self-definitional loop, fitted parameter renamed as prediction, or self-citation chain. No equations or steps in the abstract or description equate the output bound to the input approximations by construction; the framework instead provides a quantitative way to navigate tradeoffs via the Design Meta-Problem. The LQR case study instantiates the approach without forcing the general claim. The derivation is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The framework relies on standard control-theoretic assumptions rather than new postulates.

axioms (2)

domain assumption The closed-loop system satisfies incremental input-to-state stability
Invoked to bound the aggregate effect of design approximations via a sector condition.
standard math A small-gain condition can be applied to the sector-bounded difference operator
Used to conclude stability of the approximated controller.

pith-pipeline@v0.9.0 · 5465 in / 1286 out tokens · 56617 ms · 2026-05-15T06:22:28.922113+00:00 · methodology

Review history (2 revisions) →

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.

bounding the aggregate effects of these approximations reduces to verifying a design-dependent sector bound on the difference between the deployed policy and an idealized baseline, with stability enforced via a small-gain condition
IndisputableMonolith/Foundation/AlphaCoordinateFixation.lean J_uniquely_calibrated_via_higher_derivative unclear

?

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

J(ˆπN)−J(π⋆)≤(L⋆N+L⋆τ+L⋆T+L⋆M)μρ(∥x0∥)

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

25 extracted references · 25 canonical work pages

[1]

Alla and V

A. Alla and V. Simoncini. Order reduction approaches for the algebraic Riccati equation and the LQR problem. In Numerical Methods for Optimal Control Problems, pages 89--109. Springer, 2019

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[2]

D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Trans. Aut. Contr., 47 0 (3): 0 410--421, 2002

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[3]

A. C. Antoulas. Approximation of large-scale dynamical systems. SIAM, 2005

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R. R. Bitmead and M. Gevers. Riccati difference and differential equations: Convergence, monotonicity and stability. In The Riccati Equation, pages 263--291. Springer, 1991

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J. F. Bonnans and A. Festa. Error estimates for the Euler discretization of an optimal control problem with first-order state constraints. SIAM J. Numerical Analysis, 55 0 (2): 0 445--471, 2017

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[6]

Bottou and O

L. Bottou and O. Bousquet. The tradeoffs of large scale learning. Adv. Neur. Inf. Proc. Sys., 20, 2007

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[7]

Chen and B

T. Chen and B. A. Francis. Optimal sampled-data control systems. Springer-Verlag, 1995

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K. Garg, R. K. Cosner, U. Rosolia, A. D. Ames, and D. Panagou. Multi-rate control design under input constraints via fixed-time barrier functions. IEEE Contr. Sys. Lett., 6: 0 608--613, 2021

work page 2021
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L. Gr \"u ne and A. Rantzer. On the infinite horizon performance of receding horizon controllers. IEEE Trans. Aut. Contr., 53 0 (9): 0 2100--2111, 2008

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W. W. Hager and L. L. Horowitz. Convergence and stability properties of the discrete Riccati operator equation and the associated optimal control and filtering problems. SIAM J. Contr. & Optim., 14 0 (2): 0 295--312, 1976

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W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada. Event-triggered and self-triggered control. In Encyclopedia of Systems & Control, pages 724--730. Springer, 2021

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Karapetyan, E

A. Karapetyan, E. C. Balta, A. Iannelli, and J. Lygeros. Closed-loop finite-time analysis of suboptimal online control. IEEE Trans. Aut. Contr., 2025

work page 2025
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Kenney and G

C. Kenney and G. Hewer. The sensitivity of the algebraic and differential riccati equations. SIAM J. Contr. Optim., 28 0 (1): 0 50--69, 1990

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H. K. Khalil. Nonlinear Systems, volume 3. Prentice Hall, 2002

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M. M. Konstantinov, P. H. Petkov, and N. D. Christov. Perturbation analysis of the discrete Riccati equation. Kybernetika, 29 0 (1): 0 18--29, 1993

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Lahijanian, M

M. Lahijanian, M. Svorenova, A. A. Morye, B. Yeomans, D. Rao, I. Posner, P. Newman, H. Kress-Gazit, and M. Kwiatkowska. Resource-performance tradeoff analysis for mobile robots. IEEE Robot. Autom. Lett., 3 0 (3): 0 1840--1847, 2018

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Liao-McPherson, M

D. Liao-McPherson, M. M. Nicotra, and I. Kolmanovsky. Time-distributed optimization for real-time model predictive control: Stability, robustness, and constraint satisfaction. Automatica, 117: 0 108973, 2020

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Matni, A

N. Matni, A. D. Ames, and J. C. Doyle. A quantitative framework for layered multirate control: Toward a theory of control architecture. IEEE Contr. Sys. Mag., 44 0 (3), 2024

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Richter, C

S. Richter, C. N. Jones, and M. Morari. Computational complexity certification for real-time mpc with input constraints based on the fast gradient method. IEEE Trans. Aut. Contr., 57 0 (6), 2011

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Rosolia, A

U. Rosolia, A. Singletary, and A. D. Ames. Unified multirate control: From low-level actuation to high-level planning. IEEE Trans. Aut. Contr., 67 0 (12): 0 6627--6640, 2022

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Rubagotti, P

M. Rubagotti, P. Patrinos, and A. Bemporad. Stabilizing linear model predictive control under inexact numerical optimization. IEEE Trans. Aut. Contr., 59 0 (6): 0 1660--1666, 2014

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S. Shi, A. Tsiamis, and B. de Schutter. Suboptimality analysis of receding horizon quadratic control with unknown linear systems and its applications in learning-based control. IEEE Trans. Aut. Contr., 71 0 (3): 0 1422--1437, 2026

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Srikanthan, A

A. Srikanthan, A. Karapetyan, V. Kumar, and N. Matni. Closed-loop analysis of admm-based suboptimal linear model predictive control. IEEE Contr. Sys. Lett., 8: 0 3195--3200, 2024

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Stamouli, A

C. Stamouli, A. Tsiamis, M. Morari, and G. J. Pappas. Layered multirate control of constrained linear systems. In Proc. 64th IEEE Conf. Decis. Contr., pages 3027--3034, 2025

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Zardini, A

G. Zardini, A. Censi, and E. Frazzoli. Co-design of autonomous systems: From hardware selection to control synthesis. In Proc. 2021 Europ. Contr. Conf., pages 682--689, 2021

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[1] [1]

Alla and V

A. Alla and V. Simoncini. Order reduction approaches for the algebraic Riccati equation and the LQR problem. In Numerical Methods for Optimal Control Problems, pages 89--109. Springer, 2019

work page 2019

[2] [2]

D. Angeli. A Lyapunov approach to incremental stability properties. IEEE Trans. Aut. Contr., 47 0 (3): 0 410--421, 2002

work page 2002

[3] [3]

A. C. Antoulas. Approximation of large-scale dynamical systems. SIAM, 2005

work page 2005

[4] [4]

R. R. Bitmead and M. Gevers. Riccati difference and differential equations: Convergence, monotonicity and stability. In The Riccati Equation, pages 263--291. Springer, 1991

work page 1991

[5] [5]

J. F. Bonnans and A. Festa. Error estimates for the Euler discretization of an optimal control problem with first-order state constraints. SIAM J. Numerical Analysis, 55 0 (2): 0 445--471, 2017

work page 2017

[6] [6]

Bottou and O

L. Bottou and O. Bousquet. The tradeoffs of large scale learning. Adv. Neur. Inf. Proc. Sys., 20, 2007

work page 2007

[7] [7]

Chen and B

T. Chen and B. A. Francis. Optimal sampled-data control systems. Springer-Verlag, 1995

work page 1995

[8] [8]

K. Garg, R. K. Cosner, U. Rosolia, A. D. Ames, and D. Panagou. Multi-rate control design under input constraints via fixed-time barrier functions. IEEE Contr. Sys. Lett., 6: 0 608--613, 2021

work page 2021

[9] [9]

Gr \"u ne and A

L. Gr \"u ne and A. Rantzer. On the infinite horizon performance of receding horizon controllers. IEEE Trans. Aut. Contr., 53 0 (9): 0 2100--2111, 2008

work page 2008

[10] [10]

W. W. Hager and L. L. Horowitz. Convergence and stability properties of the discrete Riccati operator equation and the associated optimal control and filtering problems. SIAM J. Contr. & Optim., 14 0 (2): 0 295--312, 1976

work page 1976

[11] [11]

W. P. M. H. Heemels, K. H. Johansson, and P. Tabuada. Event-triggered and self-triggered control. In Encyclopedia of Systems & Control, pages 724--730. Springer, 2021

work page 2021

[12] [12]

Karapetyan, E

A. Karapetyan, E. C. Balta, A. Iannelli, and J. Lygeros. Closed-loop finite-time analysis of suboptimal online control. IEEE Trans. Aut. Contr., 2025

work page 2025

[13] [13]

Kenney and G

C. Kenney and G. Hewer. The sensitivity of the algebraic and differential riccati equations. SIAM J. Contr. Optim., 28 0 (1): 0 50--69, 1990

work page 1990

[14] [14]

H. K. Khalil. Nonlinear Systems, volume 3. Prentice Hall, 2002

work page 2002

[15] [15]

M. M. Konstantinov, P. H. Petkov, and N. D. Christov. Perturbation analysis of the discrete Riccati equation. Kybernetika, 29 0 (1): 0 18--29, 1993

work page 1993

[16] [16]

Lahijanian, M

M. Lahijanian, M. Svorenova, A. A. Morye, B. Yeomans, D. Rao, I. Posner, P. Newman, H. Kress-Gazit, and M. Kwiatkowska. Resource-performance tradeoff analysis for mobile robots. IEEE Robot. Autom. Lett., 3 0 (3): 0 1840--1847, 2018

work page 2018

[17] [17]

Liao-McPherson, M

D. Liao-McPherson, M. M. Nicotra, and I. Kolmanovsky. Time-distributed optimization for real-time model predictive control: Stability, robustness, and constraint satisfaction. Automatica, 117: 0 108973, 2020

work page 2020

[18] [18]

Matni, A

N. Matni, A. D. Ames, and J. C. Doyle. A quantitative framework for layered multirate control: Toward a theory of control architecture. IEEE Contr. Sys. Mag., 44 0 (3), 2024

work page 2024

[19] [19]

Richter, C

S. Richter, C. N. Jones, and M. Morari. Computational complexity certification for real-time mpc with input constraints based on the fast gradient method. IEEE Trans. Aut. Contr., 57 0 (6), 2011

work page 2011

[20] [20]

Rosolia, A

U. Rosolia, A. Singletary, and A. D. Ames. Unified multirate control: From low-level actuation to high-level planning. IEEE Trans. Aut. Contr., 67 0 (12): 0 6627--6640, 2022

work page 2022

[21] [21]

Rubagotti, P

M. Rubagotti, P. Patrinos, and A. Bemporad. Stabilizing linear model predictive control under inexact numerical optimization. IEEE Trans. Aut. Contr., 59 0 (6): 0 1660--1666, 2014

work page 2014

[22] [22]

S. Shi, A. Tsiamis, and B. de Schutter. Suboptimality analysis of receding horizon quadratic control with unknown linear systems and its applications in learning-based control. IEEE Trans. Aut. Contr., 71 0 (3): 0 1422--1437, 2026

work page 2026

[23] [23]

Srikanthan, A

A. Srikanthan, A. Karapetyan, V. Kumar, and N. Matni. Closed-loop analysis of admm-based suboptimal linear model predictive control. IEEE Contr. Sys. Lett., 8: 0 3195--3200, 2024

work page 2024

[24] [24]

Stamouli, A

C. Stamouli, A. Tsiamis, M. Morari, and G. J. Pappas. Layered multirate control of constrained linear systems. In Proc. 64th IEEE Conf. Decis. Contr., pages 3027--3034, 2025

work page 2025

[25] [25]

Zardini, A

G. Zardini, A. Censi, and E. Frazzoli. Co-design of autonomous systems: From hardware selection to control synthesis. In Proc. 2021 Europ. Contr. Conf., pages 682--689, 2021

work page 2021