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Mathematics
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Mathematical Methods for Nonlinear Control
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Mathematical Methods for Nonlinear Control

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "Engineering Mathematics".

Deadline for manuscript submissions: closed (31 January 2023) | Viewed by 8066

Special Issue Editor

Dr. Elias August

E-Mail Website
Guest Editor
Department of Engineering, Reykjavik University, Menntavegur 1, 102 Reykjavik, Iceland
Interests: control theory; systems biology; aerospace engineering; coupled nonlinear systems; stability; robustness; semidefinite programming; sum of squares decomposition

Special Issue Information

Dear Colleagues,

We invite you to submit your latest research in the area of nonlinear control theory to this Special Issue, “Mathematical Methods for Nonlinear Control”, in the journal Mathematics. Research in Nonlinear Control uses mathematical theory, mathematical modeling, and algorithms for optimal and robust control solutions in many fields. High-quality papers are solicited to address both theoretical and practical issues in the area of nonlinear control. Submissions that present new theoretical results as well as new applications are welcome. Potential topics include, but are not limited to, optimal and robust control, model predictive control, aerospace control, and automation.

Dr. Elias August
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • lyapunov theory
  • robust control
  • optimal control
  • model predictive control
  • aerospace control

Published Papers (5 papers)

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Research

14 pages, 1165 KiB  
Article
A Differential Flatness-Based Model Predictive Control Strategy for a Nonlinear Quarter-Car Active Suspension System
by Daniel Rodriguez-Guevara, Antonio Favela-Contreras, Francisco Beltran-Carbajal, Carlos Sotelo and David Sotelo
Mathematics 2023, 11(4), 1067; https://0-doi-org.brum.beds.ac.uk/10.3390/math11041067 - 20 Feb 2023
Cited by 1 | Viewed by 1391
Abstract
Controlling an automotive suspension system using an actuator is a complex nonlinear problem that requires both fast and precise solutions in order to achieve optimal performance. In this work, the nonlinear model of a quarter-car active suspension is expressed in terms of a [...] Read more.
Controlling an automotive suspension system using an actuator is a complex nonlinear problem that requires both fast and precise solutions in order to achieve optimal performance. In this work, the nonlinear model of a quarter-car active suspension is expressed in terms of a flat output and its derivatives in order to embed the nonlinearities of the system in the flat output. Afterward, a Model Predictive Controller based on the differential flatness derivation (MPC-DF) of the quarter-car is proposed in order to achieve optimal control performance in both passenger comfort and road holding without diminishing the lifespan of the wheel. This formulation results in a linear optimization problem while maintaining the nonlinear behavior of the active suspension system. Afterward, the optimization problem is solved by means of Quadratic Programming (QP), enabling real-time implementation. Simulation results are presented using a realistic road disturbance to show the effectiveness of the proposed control strategy. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Control)
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21 pages, 3927 KiB  
Article
Impact of Sequential Model Predictive Control on Induction Motor Performance: Comparison of Converter Topologies
by Duberney Murillo-Yarce, Baldomero Araya, Carlos Restrepo, Marco Rivera and Patrick Wheeler
Mathematics 2023, 11(4), 972; https://0-doi-org.brum.beds.ac.uk/10.3390/math11040972 - 14 Feb 2023
Cited by 2 | Viewed by 1396
Abstract
Finite Set Model Predictive Control (FS-MPC) is a widely used technique in power electronic converter applications. One challenge in FS-MPC implementation is selecting appropriate weighting factors, as there is currently no established methodology for finding the best values. An alternative approach is to [...] Read more.
Finite Set Model Predictive Control (FS-MPC) is a widely used technique in power electronic converter applications. One challenge in FS-MPC implementation is selecting appropriate weighting factors, as there is currently no established methodology for finding the best values. An alternative approach is to consider cost functions without weighting factors, as used by the Sequential Model Predictive Control (SMPC). In this paper, the performance of SMPC applied to induction motors is analyzed. The SMPC strategy involves sequentially evaluating simple cost functions by considering a limited number of available switching states for the power electronic converter. This number is the control parameter of the SMPC. The parameter’s domains and a selection criteria based on THD were established in this investigation. The power converter topologies studied include the Voltage Source Inverter (VSI) and the Neutral Point Clamped three-level (3L-NPC). Simulations performed in PLECS software and Hardware-in-the-Loop (HIL) tests using an RT Box for valid parameters satisfy the characteristics of the classical predictive control, such as good control variables tracking and high dynamic response. For a VSI converter, increasing the control parameter results in reduced harmonic distortion, while for an NPC converter, optimal results are achieved with control parameter values within a specific range. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Control)
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20 pages, 5287 KiB  
Article
Efficiency-Oriented MPC: Using Nested Structure to Realize Optimal Operation and Control
by Jiahong Xu and Lihong Xu
Mathematics 2022, 10(13), 2324; https://0-doi-org.brum.beds.ac.uk/10.3390/math10132324 - 02 Jul 2022
Viewed by 1303
Abstract
Optimal operation and control, which can result in the global optimal operation performance of industrial processes, has been a hot topic in recent control strategy designs. However, existing control strategies, such as real-time optimization (RTO), dynamic real-time optimization (DRTO), and economic model predictive [...] Read more.
Optimal operation and control, which can result in the global optimal operation performance of industrial processes, has been a hot topic in recent control strategy designs. However, existing control strategies, such as real-time optimization (RTO), dynamic real-time optimization (DRTO), and economic model predictive control (EMPC), have their own limitations, and they can only generate sub-optimal operation performance. In order to further improve online global operation performance, a new kind of control strategy named efficiency-oriented model predictive control (EfiMPC) is proposed in this paper. The aim of the EfiMPC is discussed first, and then, the ideal EfiMPC strategy with a nested structure is proposed, where the inner layer is the offline construction of an efficiency-oriented terminal region, and the outer layer is the direct optimization of the transient operation performance. This efficiency-oriented terminal region can guarantee a dynamic operation performance in the closed-loop perspective, and a better global operation performance can thus be obtained. A practical EfiMPC strategy, which replaces the offline construction of the efficiency-oriented terminal region with the online optimization of the average dynamic operation performance in the inner layer, is also proposed, and the recursive feasibility as well as the closed-loop stability of practical EfiMPC are discussed. Finally, a CSTR application was used to test the superiority of the proposed EfiMPC strategy, and the simulation results show that EfiMPC can obtain the best global operation performance compared with the other three control strategies; thus, the effectiveness of EfiMPC is demonstrated. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Control)
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18 pages, 19116 KiB  
Article
Toward Embedded System Resources Relaxation Based on the Event-Triggered Feedback Control Approach
by Andrej Sarjaš and Dušan Gleich
Mathematics 2022, 10(4), 550; https://0-doi-org.brum.beds.ac.uk/10.3390/math10040550 - 10 Feb 2022
Cited by 2 | Viewed by 1069
Abstract
The paper describes an event-triggered nonlinear feedback controller design. Event triggering is a real-time controller implementation technique which reduces embedded system utilization and relaxes task scheduling of the real-time system. In contrast to classic time implementation techniques, the event-triggered execution is validated regarding [...] Read more.
The paper describes an event-triggered nonlinear feedback controller design. Event triggering is a real-time controller implementation technique which reduces embedded system utilization and relaxes task scheduling of the real-time system. In contrast to classic time implementation techniques, the event-triggered execution is validated regarding the introduced triggering policy. The triggering rule is a boundary, where the last task value is preserved until the rule is violated. In the given paper, two different event-triggered strategies are designed for the class of dynamic systems with integral behavior. Both methods are based on sliding mode controller design, where the triggering rule of the first design involves only a partial state vector, which is a direct consequence of the triggering rule derivation throughout the Lyapunov stability analysis. In the second approach, the sliding mode controller is designed upon prior stabilized systems with the additional term, which enables derivation of the triggering rule based on the whole state vector. The second approach offers better closed-loop performance and higher relaxation of the system utilization. The selection of triggering boundary is related closely to the derived minimal inter-event time, which impacts the computational burden of the real-time system and closed-loop performance directly. The derived controllers are compared with the classic sample and hold implementation techniques. The real-time results are presented, and system performances are confirmed regarding embedded system task relaxation, lowering the computational intensity and preserving closed-loop dynamics. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Control)
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23 pages, 3100 KiB  
Article
Hierarchical Cognitive Control for Unknown Dynamic Systems Tracking
by Mircea-Bogdan Radac and Timotei Lala
Mathematics 2021, 9(21), 2752; https://doi.org/10.3390/math9212752 - 29 Oct 2021
Cited by 13 | Viewed by 2088
Abstract
A general control system tracking learning framework is proposed, by which an optimal learned tracking behavior called ‘primitive’ is extrapolated to new unseen trajectories without requiring relearning. This is considered intelligent behavior and strongly related to the neuro-motor cognitive control of biological (human-like) [...] Read more.
A general control system tracking learning framework is proposed, by which an optimal learned tracking behavior called ‘primitive’ is extrapolated to new unseen trajectories without requiring relearning. This is considered intelligent behavior and strongly related to the neuro-motor cognitive control of biological (human-like) systems that deliver suboptimal executions for tasks outside of their current knowledge base, by using previously memorized experience. However, biological systems do not solve explicit mathematical equations for solving learning and prediction tasks. This stimulates the proposed hierarchical cognitive-like learning framework, based on state-of-the-art model-free control: (1) at the low-level L1, an approximated iterative Value Iteration for linearizing the closed-loop system (CLS) behavior by a linear reference model output tracking is first employed; (2) an experiment-driven Iterative Learning Control (EDILC) applied to the CLS from the reference input to the controlled output learns simple tracking tasks called ‘primitives’ in the secondary L2 level, and (3) the tertiary level L3 extrapolates the primitives’ optimal tracking behavior to new tracking tasks without trial-based relearning. The learning framework relies only on input-output system data to build a virtual state space representation of the underlying controlled system that is assumed to be observable. It has been shown to be effective by experimental validation on a representative, coupled, nonlinear, multivariable real-world system. Able to cope with new unseen scenarios in an optimal fashion, the hierarchical learning framework is an advance toward cognitive control systems. Full article
(This article belongs to the Special Issue Mathematical Methods for Nonlinear Control)
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