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REVIEW 4 major objections 5 minor 35 references

Iteration-Free Cooperative Distributed MPC through Multiparametric Programming

T0 review · 4 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read This paper proposes replacing the iterative information exchange in cooperative distributed MPC with simultaneous solution of offline-computed explicit control laws, cutting communication to one exchange per sample step while preserving…

desk verdict A genuinely new simultaneous-solve idea for iteration-free DiMPC, but the equivalence to the iterative solution is asserted rather than proved, the computational comparison is overstated, and Algorithm 2 can pick spurious solutions. read the letter →

arxiv 2411.14319 v2 pith:QUO4QAS2 submitted 2024-11-21 eess.SY cs.SY

Radhe S. T. Saini , Parth R. Brahmbhatt , Styliani Avraamidou , Hari S. Ganesh This is my paper · ORCID
classification eess.SYcs.SY
keywords cooperativedistributedMPCmultiparametricprogrammingexplicitcontrollawscriticalregionsiteration-freecommunicationloadlinearsubsystems
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 tries to remove the iterative information-exchange loop from cooperative distributed model predictive control by replacing online iterations with offline-computed explicit control laws that are solved simultaneously. If the approach works, each local controller exchanges information only once per sampling period instead of dozens or hundreds of times, reducing latency and network load while preserving centralized-like performance. The numerical case studies with two to five coupled linear subsystems support the claim mainly for the V2 variant, whose computation time is below that of the iterative multiparametric version for all tested subsystem counts. The authors argue that because the reformulation does not change the solution space, the stability properties of the iterative algorithm carry over without a new proof.

What carries the argument

The load-bearing object is the explicit multiparametric control law: for each local controller i, the optimal input trajectory is written as Ui = f_i^v(θi) whenever the parameter vector θi = (x(k), U1, ..., Ui-1, Ui+1, ..., UM) lies in critical region CR_i^v, a polyhedron defined by Φ_i^v θi ≤ φ_i^v. By splitting θi into the measured state and the other controllers' input trajectories, the local laws become simultaneous equations in the full input vector U, parametrized only by the state x(k); the algorithms solve these equations jointly rather than iterating. V2 exploits the observation that states change gradually, restricting the simultaneous solve to the previous optimal critical region and its neighbors and falling back to iterative I-mpDiMPC if no solution is found, which bounds worst-case execution time by the iterative algorithm's.

What would settle it

Compare the closed-loop input sequence of IF-mpDiMPC-V2 with the converged iterative DiMPC solution using the paper's 1e-8 tolerance on a coupled two-subsystem plant whose optimal critical region jumps between distant regions in successive sample times; if the inputs diverge at any sample, or if V2's fallback to the iterative algorithm triggers, then the iteration-free solve is not equivalent to the converged solution or the one-exchange-per-step bound is not achieved in that step.

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Extended reading notes

Core claim

The central claim is that cooperative DiMPC can be made iteration-free by computing, offline, an explicit multiparametric solution for each local controller: an affine function of the state and of the other controllers' input trajectories, valid in a polyhedral critical region. Online, the local control-law equations are assembled and solved simultaneously for the full input vector instead of being iterated to consensus. Three variants are proposed: a direct simultaneous solve over all critical-region combinations (IF-mpDiMPC), a variant that first prunes infeasible critical regions via linear programming (IF-mpDiMPC-V1.5), and a variant that searches only the critical region found at the previous sample time plus its neighbors (IF-mpDiMPC-V2). The numerical results show V2 is the fastest of the three and faster than iterative multiparametric DiMPC for two, three, four, and five subsystems, while all three reduce data transfers to one per time step. The paper asserts that the simultaneous formulation does not alter the solution space, so the iterative algorithm's stability guarantees apply to the iteration-free versions.

Load-bearing premise

The whole approach rests on the assumption that solving the local explicit control laws simultaneously gives exactly the inputs the converged iterative cooperative DiMPC would give, so the iterative method's stability guarantee transfers without a new proof.

Editorial extensions

If this is right

  • Data transfer between local controllers drops from tens to a hundred exchanges per sample time to exactly one, removing iteration-dependent latency from the control loop.
  • IF-mpDiMPC-V2 reported average computation time below that of iterative I-mpDiMPC for two, three, four, and five subsystems in the random-plant case studies.
  • Because all control laws are generated offline, online effort reduces to critical-region search and algebraic evaluation, making real-time computation more predictable.
  • If the simultaneous solve is equivalent to the converged cooperative solution, the closed-loop behavior is centralized-like without the communication cost of reaching consensus.
  • The fallback in IF-mpDiMPC-V2 guarantees that cases missed by the neighbor search still terminate within the time bound of the iterative algorithm.

Reading between the lines

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

  • The exponential growth of critical-region combinations suggests that IF-mpDiMPC and IF-mpDiMPC-V1.5 will not scale to larger plants; the reported advantage of V2 depends on the heuristic that states and optimal regions move gradually, which the paper does not test under abrupt disturbances.
  • The one-exchange-per-step communication count is nominal: the paper does not model transmission delay, packet loss, or asynchronous arrival, so the latency and robustness benefits are a qualitative architectural consequence rather than a measured quantity.
  • A testable extension is to generate the offline laws robustly by treating bounded state variation as an additional parameter, and to check whether V2's neighbor search plus fallback preserves the equivalence claim when the optimal critical region jumps between distant regions.
  • The equivalence argument would be stronger if the paper showed uniqueness of the simultaneous solution across critical-region combinations, since multiple feasible combinations would require the added selection rule of choosing the least objective value.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 5 minor

Summary. The paper proposes replacing the iterative information exchange of cooperative distributed MPC (DiMPC) with a single simultaneous solution of offline-computed multiparametric (explicit) control laws. Three variants are introduced: IF-mpDiMPC, which solves the coupled explicit laws for all critical-region combinations; IF-mpDiMPC-V1.5, which pre-filters regions by a feasibility LP; and IF-mpDiMPC-V2, which restricts candidate regions to the previous sample's region and its neighbors. The authors claim these methods reduce both communication load and online computation time relative to iterative DiMPC and iterative mpDiMPC (I-mpDiMPC), and they report closed-loop simulations on random linear plants with M = 2 to 5 coupled subsystems, showing centralized-like tracking performance.

Significance. The core idea is attractive and potentially useful: if the simultaneous solve of the local explicit laws provably reproduces the converged cooperative DiMPC solution, then cooperative DiMPC can be implemented with a single communication round and fixed latency on small linear plants. The offline explicit laws are derived from problem data without parameter fitting, and the comparisons use the same tuning across methods, so the experimental design is fair. However, the paper's central claims are currently supported only by assertion, not proof, and the numerical data in Table 1 partially contradict the stated computational advantages. The contribution would be significant if the equivalence and stability inheritance were rigorously established and the claims corrected to match the data.

major comments (4)
  1. [Section 3.3, Algorithms 2-4, Eq. (18)] The statement that 'the reformulation does not alter the underlying solution space or the optimal solutions obtained through multiparametric programming' is asserted without proof, and the inherited-stability conclusion rests entirely on it. In Eq. (18), the law U_i = f_i^v(θbar, V_i) is valid only when (θbar, V_i) lies in CR_i^v. In the simultaneous solve, V_i is part of the unknown U, so the validity of a tuple's laws must be verified a posteriori by checking the returned U against the tuple's region inequalities. Algorithm 2 solves the linear equations for every tuple but never performs that verification. Selecting the lowest-OCP objective among solutions that may come from inactive tuples can return a point that is not a fixed point of the I-mpDiMPC iteration, so the closed-loop input may differ from I-mpDiMPC and the stability argument collapses. A proof or a concrete verification step is required before the equivalence and stability claims can be accepted.
  2. [Section 5, Table 1 and text; Section 6] The claim that 'the computation time of IF-mpDiMPC-V2 is less than that of I-mpDiMPC, again for all number of subsystem cases' is directly contradicted by Table 1 for M=2: IF-mpDiMPC-V2 takes 0.205 s versus 0.074 s for I-mpDiMPC. The conclusion's stronger wording—'performed the best in terms of the computation time among all the controllers and for all the number of subsystem cases'—is thus false. Additionally, the abstract and introduction state that the proposed methods 'significantly improve the computational cost' compared to DiMPC, but Table 1 shows IF-mpDiMPC at M=3 takes 167.218 s versus 24.884 s for DiMPC, i.e., it is much slower. These claims must be corrected and the comparison restated with the M=2 and M=3 exceptions made explicit.
  3. [Section 3.2, Eq. (24)] The feasibility check in IF-mpDiMPC-V1.5 is only a pre-filter on the existence of some V_i satisfying the region inequalities, not a verification that the solution produced by the simultaneous equations actually lies in the candidate regions. The LP in Eq. (24) minimizes zero, so it only returns feasibility information; a region can pass the check while the subsequently computed U violates that same region's inequalities. Therefore the reduction from n_CR,i to n*_CR,i does not by itself guarantee that all tuples retained are valid, and the candidate-selection mechanism remains incomplete unless an explicit region-membership test is added after the linear solve.
  4. [Section 3.1, Eqs. (19)-(20)] No conditions are stated for the simultaneous linear systems to be nonsingular or to have a unique solution. For a given tuple of critical regions, the equations U = g_i^v(θbar) may be singular or may admit multiple solutions, and the algorithms do not specify how such cases are handled. The selection rule 'the one that results in the least value of the objective' is meaningful only when at least one valid solution exists and is not well-defined for a singular system with infinitely many solutions. The paper should either state the regularity assumptions (e.g., full column rank of the resulting linear map) or provide a fallback that covers singular and underdetermined cases.
minor comments (5)
  1. [Eq. (19), M-th row] The last condition in the assembled system uses 'Φ^v_{i,2}' instead of 'Φ^v_{M,2}', which is an inconsistency in the notation for the M-th controller's critical-region inequality.
  2. [Sections 2.3 and 3, Algorithm 1] The name 'Wegenstein' is a misspelling; the correct reference is Wegstein [32]. This appears in Algorithm 1 and in the text describing the accelerated convergence procedure.
  3. [Eq. (13) and Algorithm 1] The overbar notation for the warm-start vector and the iterative update is used inconsistently: U(p) and ¯U(p) alternate without a clear distinction, which makes it hard to follow which quantity is transmitted and which one is used in the convergence check. Please define both notations precisely in one place.
  4. [Section 5, paragraph after Figure S4] The text says 'The mean computation time for various control architectures is shown in Table 1' but does not define what the mean is taken over; state that it is the average over the 100 random plants per subsystem count and over the closed-loop horizon, or clarify the averaging procedure.
  5. [Section 5, Figure S4 reference] The reference to 'Figures S4' should be 'Figure S4' (the computation-time figure), and the supplementary figures S5-S7 are not explicitly referenced in the main text; consider adding in-text references or removing the orphan figure numbers.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the IF-mpDiMPC laws are derived from the OCP and compared against independent iterative baselines; the unproven equivalence and stability-inheritance assertion in Section 3.3 is a support gap, not a definitional or fitted-input reduction.

full rationale

The paper's derivation chain is not circular. The multiparametric control laws (eqs. 16-18) are generated offline from the local OCPs (eq. 15), with no parameter fitted to enforce agreement with the iterative DiMPC solution. The iteration-free algorithms replace iteration by simultaneous solution of the explicit affine laws; the output is not defined to be the DiMPC fixed point, and the paper reports computation-time differences (Table 1) and admits fallback to iterative I-mpDiMPC when V2 finds no solution, so the comparison is externally meaningful rather than forced by construction. Self-citations [26,27] appear only as background for multiparametric DiMPC computation-time reduction and are not load-bearing for the novelty claim. The closest-to-circular element is Section 3.3's statement that 'the reformulation does not alter the underlying solution space or the optimal solutions obtained through multiparametric programming. Hence, the stability properties of the iterative algorithms are also applicable'—this is an asserted equivalence, not a reduction by construction, and the conditional validity of the local laws (eq. 18) is not verified in Algorithm 2. However, that is a correctness or rigor concern, not a circularity pattern: the simulation outcomes are not forced by a fitted input or by a self-citation chain. The paper's own stated limitation that worst-case scenarios and robustness were not explored further confirms that the claims are empirical rather than tautological.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central method inherits nearly all of its analytic content from offline mpQP theory and from the iterative DiMPC baseline; the paper contributes the simultaneous-solve organization and region pruning. It introduces no fitted calibration beyond simulation tuning, but the equivalence and stability inheritance is an unproved assumption, and offline mpQP partition correctness is taken from POP Toolbox. The numerical conclusions also depend on several unreported tunable parameters such as Q, R, P, rho, wmin, and wmax.

free parameters (4)
  • Prediction horizon Np = 3
    Set to 3 for all case studies; the number of critical regions and the size of the simultaneous equation system grow with Np, so this choice bounds the claimed computational results.
  • Objective weights Q, R, P and subsystem weights rho_i = not reported
    Values are never stated; they define the plantwide cost and the mp control laws, so closed-loop performance and region geometry depend on them.
  • Wegstein acceleration bounds wmin, wmax = not reported
    Algorithm 1 clamps w with wmin and wmax but the paper gives no values; these affect convergence of the DiMPC and I-mpDiMPC baselines and thus the comparison.
  • Convergence tolerance epsilon and iteration cap pmax = 1e-8 and 100
    Chosen to make iterative baselines approach centralized performance, but Table 2 shows pmax=100 is reached for M=4 and M=5, so the baselines may not actually have converged.
assumptions (4)
  • domain assumption The plantwide OCP is convex and has a unique fixed point under coordinate-wise minimization, so iterating and solving simultaneously give the same solution.
    No convexity or uniqueness conditions are stated. The equivalence and inherited stability claims in Section 3.3 depend on this.
  • domain assumption POP Toolbox returns a complete and correct multiparametric QP solution for each local OCP.
    All online results inherit correctness and completeness from the offline mp solver; no verification of the partition is provided.
  • ad hoc to paper The reformulation does not alter the underlying solution space, and stability is inherited from iterative DiMPC.
    This is asserted in the last paragraph of Section 3.3 without proof and is the main load-bearing premise.
  • domain assumption The random plant model exactly represents the true plant, with no disturbances or model mismatch.
    Simulations use the same model for prediction and plant, so robustness to mismatch is out of scope.

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Cite this review

Pith. "Pith review of Iteration-Free Cooperative Distributed MPC through Multiparametric Programming." pith.science (2026). https://pith.science/paper/QUO4QAS2

@misc{pith2026241114319,
  author       = {Pith},
  title        = {Pith review of: Iteration-Free Cooperative Distributed MPC through Multiparametric Programming},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/QUO4QAS2}},
  note         = {Machine review of arXiv:2411.14319}
}
read the original abstract

Cooperative Distributed Model Predictive Control (DiMPC) architecture employs local MPC controllers to control different subsystems, exchanging information with each other through an iterative procedure to enhance overall control performance compared to the decentralized architecture. However, this method can result in high communication between the controllers and computational costs. In this work, the amount of information exchanged and the computational costs of DiMPC are reduced significantly by developing novel iteration-free solution algorithms based on multiparametric (mp) programming. These algorithms replace the iterative procedure with simultaneous solutions of explicit mpDiMPC control law functions. The reduced communication among local controllers decreases system latency, which is crucial for real-time control applications. The effectiveness of the proposed iteration-free mpDiMPC algorithms is demonstrated through comprehensive numerical simulations involving groups of coupled linear subsystems, which are interconnected through their inputs and a cooperative plant-wide cost function.

Figures

Figures reproduced from arXiv: 2411.14319 by the authors.

Figure 1
Figure 1. The number of critical regions for different number of subsystems on a log scale. Note that the [PITH_FULL_IMAGE:figures/full_fig_p023_1.png] view at source ↗
Figure 2
Figure 2. Controller performance for 2 subsystems. (a) Outputs of each subsystem. (b) Applied inputs to [PITH_FULL_IMAGE:figures/full_fig_p024_2.png] view at source ↗
Figure 3
Figure 3. Computation times in log scale for the different DiMPC control architectures. [PITH_FULL_IMAGE:figures/full_fig_p025_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Number of instances of data transfer in log scale for the different DiMPC control architectures for [PITH_FULL_IMAGE:figures/full_fig_p027_4.png]

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Works this paper leans on

35 extracted references · 34 canonical work pages

  1. [1]

    C. R. Cutler, B. L. Ramaker, Dynamic matrix control?? a computer control algorithm, in: joint automatic control conference, 17, 1980, p. 72

    work page 1980

  2. [2]

    S. J. Qin, T. A. Badgwell, An overview of industrial model predictive control technology, in: AIche symposium series, volume 93, New York, NY: American Institute of Chemical Engineers, 1971-c2002., 1997, pp. 232–256

    work page 1971

  3. [3]

    Morari, J

    M. Morari, J. H. Lee, Model predictive control: past, present and future, Computers & Chemical Engineering 23 (1999) 667–682

    work page 1999

  4. [4]

    J. B. Rawlings, D. Q. Mayne, M. Diehl, Model predictive control: theory, computation, and design, volume 2, Nob Hill Publishing Madison, WI, 2017

    2017

  5. [5]

    Maciejowski, Predictive Control: With Constraints, Prentice Hall, 2002

    J. Maciejowski, Predictive Control: With Constraints, Prentice Hall, 2002

    work page 2002

  6. [6]

    P. A. Trodden, J. M. Maestre, Distributed predictive control with minimization of mutual disturbances, Automatica 77 (2017) 31–43

    work page 2017

  7. [7]

    H. Cui, E. W. Jacobsen, Performance limitations in decentralized control, Journal of Process Control 12 (2002) 485–494

    work page 2002

  8. [8]

    D. D. Siljak, Decentralized control of complex systems, Academic Press, New York, USA, 1991

    work page 1991

Show all 35 references
  1. [9]

    R. R. Negenborn, J. M. Maestre, Distributed model predictive control: An overview and roadmap of future research opportunities, IEEE Control Systems Magazine 34 (2014) 87–97. 28

    2014

  2. [10]

    P. D. Christofides, R. Scattolini, D. M. de la Pena, J. Liu, Distributed model predic- tive control: A tutorial review and future research directions, Computers & Chemical Engineering 51 (2013) 21–41

    2013

  3. [11]

    Avraamidou, E

    S. Avraamidou, E. N. Pistikopoulos, A multi-parametric bi-level optimization strategy for hierarchical model predictive control, in: Computer aided chemical engineering, volume 40, Elsevier, 2017, pp. 1591–1596

    2017

  4. [12]

    Richards, J

    A. Richards, J. P. How, Robust distributed model predictive control, International Journal of control 80 (2007) 1517–1531

    2007

  5. [13]

    A. N. Venkat, J. B. Rawlings, S. J. Wright, Distributed model predictive control of large-scale systems, in: Assessment and Future Directions of Nonlinear Model Predictive Control, Springer, 2007, pp. 591–605

    2007

  6. [14]

    Farina, R

    M. Farina, R. Scattolini, Distributed predictive control: A non-cooperative algorithm with neighbor-to-neighbor communication for linear systems, Automatica 48 (2012) 1088–1096

    2012

  7. [15]

    Ferramosca, D

    A. Ferramosca, D. Lim´ on, I. Alvarado, E. F. Camacho, Cooperative distributed mpc for tracking, Automatica 49 (2013) 906–914

    2013

  8. [16]

    B. T. Stewart, A. N. Venkat, J. B. Rawlings, S. J. Wright, G. Pannocchia, Cooperative distributed model predictive control, Systems & Control Letters 59 (2010) 460–469

    2010

  9. [17]

    J. Wang, Y. Yang, An improved iterative solution for cooperative distributed mpc, Automatica 140 (2022) 110155

    2022

  10. [18]

    Darivianakis, A

    G. Darivianakis, A. Eichler, J. Lygeros, Distributed model predictive control for linear systems with adaptive terminal sets, IEEE Transactions on Automatic Control 65 (2019) 1044–1056

    2019

  11. [19]

    L. Wu, X. Yin, L. Pan, J. Liu, Distributed economic predictive control of integrated en- ergy systems for enhanced synergy and grid response: A decomposition and cooperation strategy, Applied Energy 349 (2023) 121627

    2023

  12. [20]

    Y. Feng, S. Yu, H. Chen, Y. Li, S. Shi, J. Yu, H. Chen, Distributed mpc of vehicle platoons with guaranteed consensus and string stability, Scientific Reports 13 (2023) 10396

    2023

  13. [21]

    Langner, W

    F. Langner, W. Wang, M. Frahm, V. Hagenmeyer, Model predictive control of dis- tributed energy resources in residential buildings considering forecast uncertainties, En- ergy and Buildings 303 (2024) 113753. 29

    2024

  14. [22]

    P. Lu, N. Zhang, L. Ye, E. Du, C. Kang, Advances in model predictive control for large- scale wind power integration in power systems: A comprehensive review, Advances in Applied Energy (2024) 100177

    2024

  15. [23]

    Z. Zhu, G. Guan, K. Wang, Distributed model predictive control based on the al- ternating direction method of multipliers for branching open canal irrigation systems, Agricultural Water Management 285 (2023) 108372

    2023

  16. [24]

    E. N. Pistikopoulos, V. Dua, N. A. Bozinis, A. Bemporad, M. Morari, On-line optimiza- tion via off-line parametric optimization tools, Computers & Chemical Engineering 26 (2002) 175–185

    2002

  17. [25]

    Bemporad, M

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

    2002

  18. [26]

    R. S. Saini, I. Pappas, S. Avraamidou, H. S. Ganesh, Noncooperative distributed model predictive control: A multiparametric programming approach, Industrial & Engineering Chemistry Research (2023)

    2023

  19. [27]

    N. A. Diangelakis, S. Avraamidou, E. N. Pistikopoulos, Decentralized multiparametric model predictive control for domestic combined heat and power systems, Ind. Eng. Chem. Res. 55 (2016) 3313–3326

    2016

  20. [28]

    Kirubakaran, T

    V. Kirubakaran, T. Radhakrishnan, N. Sivakumaran, Distributed multiparametric model predictive control design for a quadruple tank process, Measurement 47 (2014) 841–854

    2014

  21. [29]

    Camponogara, D

    E. Camponogara, D. Jia, B. H. Krogh, S. Talukdar, Distributed model predictive control, IEEE control systems magazine 22 (2002) 44–52

    2002

  22. [30]

    Chen, et al., Improved distributed model predictive control with control planning set, Journal of Control Science and Engineering 2016 (2016)

    W. Chen, et al., Improved distributed model predictive control with control planning set, Journal of Control Science and Engineering 2016 (2016)

    2016

  23. [31]

    Farina, R

    M. Farina, R. Scattolini, Distributed non-cooperative mpc with neighbor-to-neighbor communication, IF AC Proceedings Volumes 44 (2011) 404–409

    2011

  24. [32]

    J. H. Wegstein, Accelerating convergence of iterative processes, Communications of the ACM 1 (1958) 9–13

    1958

  25. [33]

    I. I. Cplex, V12. 1: User’s manual for cplex, International Business Machines Corpora- tion 46 (2009) 157. 30

    2009

  26. [34]

    Oberdieck, N

    R. Oberdieck, N. A. Diangelakis, M. M. Papathanasiou, I. Nascu, E. N. Pistikopoulos, Pop–parametric optimization toolbox, Industrial & Engineering Chemistry Research 55 (2016) 8979–8991. 31 Supplemental Material: Iteration-Free Cooperative Distributed MPC through Multiparametr...

    2016

  27. [35]

    All figures are labeled and described in detail below

    Supporting Information This section contains additional figures that support the analysis presented in the main manuscript. All figures are labeled and described in detail below. (a) (b) Figure S1: Controller performance for 3 subsystems. (a) Outputs of each subsystem. (b) App...

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