model predictive control phd positions

• Times Higher Education
• Posted on: 11 July 2024

PhD Candidate in Real Implementation of Model Predictive Control for Building Heating

Job information, offer description.

About the job

Buildings are energy-flexible in the sense that they can move their loads in time, typically to provide services to the electricity or district heating grids (also called demand response). For instance, one key objective can be to decrease energy use during peak hours while maintaining acceptable comfort for occupants. In cold climates, the space-heating is a dominant load and advanced control can adapt this load by changing the indoor temperature in time. Model Predictive Control (MPC) gets increasing interest in performing this control. While this solution has been extensively investigated using simulations (meaning virtual experiments), there are still limited examples where MPC has been deployed and tested in a real building and over a long period. However, real buildings can differ significantly from simulations. Firstly, the amount of measurement data can be limited, and not properly structured or standardized. Secondly, the building may not be built and operated exactly as originally planned. Thirdly, physical phenomena may be more complex in reality than in simulation, for instance regarding the heating system and the occupant behavior. Finally, the existing “legacy” automation system may not be suited. Therefore, this PhD aims to define an MPC setup (or technical pathway) that can support some of these limitations and to test it in a real non-residential building. The final objective is to provide thorough documentation of the test case.

The PhD thesis is part of  Green2050, the Centre for Green Shift in the Built Environment . This centre aims to be an arena of networking and collaboration between academia and the industry. A hub for accelerating existing, planned, and new projects. The goal is to transition into a carbon-neutral built environment by 2050.

For a position as a PhD Candidate, the goal is a completed doctoral education up to an obtained doctoral degree.

Your immediate leader is Laurent Georges, Professor NTNU. The co-supervisor is Dr. John Clauss from SINTEF Community.

Duties of the position

• Define and develop an MPC setup (including the controller) that is suited for typical waterborne heating systems in non-residential buildings (such as schools or offices).  
• Test the MPC in one real building located in Trondheim and monitor the performance of the controller.
• Document in detail the performance of the MPC setup and summarize the lessons learned.

Required selection criteria

• You must have a professionally relevant background in control, building, or HVAC engineering. A background in applied mathematics can also be relevant if there is a strong focus on data-driven modeling, machine learning, and control.
• Your education must correspond to a five-year Norwegian degree program, where 120 credits are obtained at master's level
• You must have a strong academic background from your previous studies and an average grade from the master's degree program, or equivalent education, which is equal to B or better compared with NTNU's grading scale. If you do not have letter grades from previous studies, you must have an equally good academic basis. If you have a weaker grade background, you may be assessed if you can document that you are particularly suitable for a PhD education.
• Master's students can apply, but the master's degree must be obtained and documented.
• You must meet the requirements for admission to the faculty's doctoral program,  Doctoral Programme - PhD - Faculty of Engineering Science - NTNU

The appointment is to be made in accordance with  Regulations on terms of employment for positions such as postdoctoral fellow, Phd candidate, research assistant and specialist candidate  and  Regulations concerning the degrees of Philosophiae Doctor (PhD) and Philosodophiae Doctor (PhD) in artistic research national guidelines for appointment as PhD, post doctor and research assistant 

Preferred selection criteria

• Previous experience and strong interest in control and data-driven modeling. For instance, a previous experience with MPC, ideally applied to buildings (either using simulations or real cases).
• Knowledge or previous experience with building automation is a plus. 
• Excellent background in programming (ex. Python/Julia/Matlab).
• Previous experience in scientific writing (such as a journal article or conference paper). 
• Good written and oral English. Knowledge of the Norwegian language is considered as an advantage.

Personal characteristics

• Able to work in a team and follow common instructions.
• Creative, pragmatic, and solution-oriented.
• Independent, able to organize the research work and to take initiative.

Emphasis will be placed on personal and interpersonal qualities.

• exciting and stimulating tasks in a strong international academic environment
• an open and  inclusive work environment  with dedicated colleagues
• favourable terms in the  Norwegian Public Service Pension Fund
• employee benefits

Salary and conditions

As a PhD candidate (code 1017) you are normally paid from gross NOK 532 200 per annum before tax, depending on qualifications and seniority. From the salary, 2% is deducted as a contribution to the Norwegian Public Service Pension Fund.

The period of employment is 3 years. Appointment to a PhD position requires that you are admitted to the PhD programme in Engineering,  Doctoral Programme - PhD - Faculty of Engineering Science - NTNU  within three months of employment, and that you participate in an organized PhD programme during the employment period.

The engagement is to be made in accordance with the regulations in force concerning  State Employees and Civil Servants , and the acts relating to Control of the Export of Strategic Goods, Services and Technology. Candidates who by assessment of the application and attachment are seen to conflict with the criteria in the latter law will be prohibited from recruitment to NTNU. After the appointment you must assume that there may be changes in the area of work.

It is a prerequisite you can be present at and accessible to the institution daily.

About the application

The application and supporting documentation to be used as the basis for the assessment must be in English.

Publications and other scientific work must be attached to the application. Please note that your application will be considered based solely on information submitted by the application deadline. You must therefore ensure that your application clearly demonstrates how your skills and experience fulfil the criteria specified above.

The application must include: 

• CV and certificates
• A cover letter where your motivation to follow a PhD education, your interest, and your background for this specific PhD topic should be clearly explained.
• Transcripts and diplomas for bachelor's and master's degrees. If you have not completed the master's degree, you must submit a confirmation that the master's thesis has been submitted.
• A copy of the master's thesis. If you recently have submitted your master's thesis, you can attach a draft of the thesis. Documentation of a completed master's degree must be presented before taking up the position.
• Project proposal: Candidates are required to submit a sketch (maximum two pages) on how they plan to use their scientific background and research ambitions to address the research questions of this PhD position.
• Name and contact information of three referees
• If you have publications or other relevant research work. For each of them, describe your own contribution to the paper and the contributions/impact to the field.

If all, or parts, of your education has been taken abroad, we also ask you to attach documentation of the scope and quality of your entire education, both bachelor's and master's education, in addition to other higher education. Description of the documentation required can be found  here . If you already have a statement from  Norwegian Directorate for Higher Education and Skills , please attach this as well.

We will take joint work into account. If it is difficult to identify your efforts in the joint work, you must enclose a short description of your participation.

In the evaluation of which candidate is best qualified, emphasis will be placed on education, experience and personal and interpersonal qualities. Motivation, ambitions, and potential will also count in the assessment of the candidates. 

NTNU is committed to following evaluation criteria for research quality according to  The San Francisco Declaration on Research Assessment - DORA.

General information

Working at NTNU

NTNU believes that inclusion and diversity is our strength. We want to recruit people with different competencies, educational backgrounds, life experiences and perspectives to contribute to solving our social responsibilities within education and research. We will facilitate for our employees’ needs.

NTNU is working actively to increase the number of women employed in scientific positions and has a number of resources to  promote equality.

Department of Energy and Process Engineering has established  EPT Women in Science . The group is focused on supporting female PhD Candidates, Postdoctoral Fellows, Research Assistants and permanent academic employees within the Department. This support aims to help develop the careers of female PhD Candidates, Postdocs and Research Assistants, and is also made visible to our student body to encourage them to consider an academic path. As part of the EPT Women in Science initiative we are building an international network, inviting prominent female academics within and beyond the field of Engineering to speak at our events.

The city of Trondheim  is a modern European city with a rich cultural scene. Trondheim is the innovation capital of Norway with a population of 200,000. The Norwegian welfare state, including healthcare, schools, kindergartens and overall equality, is probably the best of its kind in the world. Professional subsidized day-care for children is easily available. Furthermore, Trondheim offers great opportunities for education (including international schools) and possibilities to enjoy nature, culture and family life and has low crime rates and clean air quality.

As an employee at NTNU, you must at all times adhere to the changes that the development in the subject entails and the organizational changes that are adopted.

A public list of applicants with name, age, job title and municipality of residence is prepared after the application deadline. If you want to reserve yourself from entry on the public applicant list, this must be justified. Assessment will be made in accordance with  current legislation . You will be notified if the reservation is not accepted.

If you have any questions about the position, please contact Laurent Georges, telephone +47 73592484, email:  [email protected]  . If you have any questions about the recruitment process, please contact Ingrid Wiggen, e-mail:  [email protected] .

If you think this looks interesting and in line with your qualifications, please submit your application electronically via jobbnorge.no with your CV, diplomas and certificates attached. Applications submitted elsewhere will not be considered. Upon request, you must be able to obtain certified copies of your documentation.  

Application deadline: 15.08.24

NTNU - knowledge for a better wo rld

The Norwegian University of Science and Technology (NTNU) creates knowledge for a better world and solutions that can change everyday life.

Department of Energy and Process Engineering

We conduct research and teaching covering the entire energy chain, from resources to the end-user. We look at how energy is produced and used by humans and machines in a sustainable way with regard to health, climate change and the resource base.  The Department of Energy and Process Engineering  is one of eight departments in the  Faculty of Engineering.

Questions about the position

Laurent Georges Professor [email protected]

Deadline  15th August 2024 Employer  NTNU - Norwegian University of Science and Technology Municipality  Trondheim Scope  Fulltime Duration Fixed Term Place of service  Høgskoleringen 1, 7491 Trondheim

Where to apply

Requirements, additional information, work location(s), share this page.

• Application
• Data Science Colloquia

Model Predictive Control

Provided by: , from: , sede: , lecturers: , semester: , hours: , exam: , educational goals: , prerequisites: , programme: , in evidence, academic year 2021-2022 (37th cycle).

As the Italian Ministry has decided to invest in a doctoral program on Artificial Intelligence, next year the Data Science Ph.D. will become one of the 5 nodes of a new national initiative: the National Artificial Intelligence Ph.D. All the partner institutions of the current program will join the new Ph.D.

The call for admissions to the National PhD in Artificial Intelligence is now open!

Interested in a multi-disciplinary PhD course oriented at cutting-edge research in human-centered Artificial Intelligence and its impacts on society? Apply to one of 44 fully-funded positions at the National PhD in AI – “Society” area:

https://dottorato.unipi.it/index.php/en/application-process-for-the-acad...

Deadline: July 23, 2021, h 13:00 CET

The program is launched by the University of Pisa in partnership with:

- National Research Council - CNR - Scuola Superiore Sant’Anna - Scuola Normale Superiore - Scuola IMT Lucca - Università di Firenze - Università di Modena e Reggio Emilia - Università di Siena - Università di Trento

and in collaboration with:

- Università di Bari - Università di Bologna - Università Cattolica del Sacro Cuore - Università dell’Aquila - Università degli Studi di Napoli L’Orientale - Università di Sassari - Università di Trieste - INDAM (Istituto Nazionale di Alta Matematica “Francesco Severi”) - Open Fiber SpA

This opportunity is part of the Italian National PhD Program in Artificial Intelligence. Overall, PhD-AI.it is made of 5 federated PhD courses that bring together 61 Italian universities and research institutions. The 5 PhD courses share a common basis in the foundations and developments of AI, and each one has an area of specialisation in a strategic sector of AI application. Each PhD course is organized by a lead university, in collaboration with the National Research Council CNR:

- Health and life sciences, Università Campus Bio-Medico di Roma - Agrifood and environment, Università degli Studi di Napoli Federico II - Security and cybersecurity, Sapienza Università di Roma - Industry 4.0, Politecnico di Torino - Society, Università di Pisa

Link to the calls for admissions to all the 5 PhD course are available at http://www.PhD-AI.it

Site created and managed by Vittorio Romano

PhD position in Model Predictive Control, Leibniz University Hannover, Germany

Mueller-irt.

We offer one PhD position at the Institute of Automatic Control at the Leibniz University Hannover, Germany, in the area Model Predictive Control. The scope of work mainly includes research activities within a research project in cooperation with a company from the area of automated test tools. The project is dedicated to the use of Model Predictive Control (MPC) methods for the implementation of test scenarios in the field of autonomous driving. Within the scope of these activities, various existing MPC methods will be adapted and implemented, and also new methods will be developed for problems arising from the considered application, such as robustness and real-time capability.

We offer a competitive salary according to the German pay scale TVL-13, including social benefits. The candidate is expected to hold a Master degree in control engineering or a related subject with specialization in control. Experience in optimization-based control (model predictive control) would be desirable. Also, teaching assistance in bachelor and master level control courses is expected.

Leibniz University Hannover considers itself a family-friendly university and therefore promotes a balance between work and family responsibilities. Part-time employment can be arranged on request, as long as the offered workplace is covered in full extent. The university aims to promote equality between women and men. For this purpose, the university strives to reduce under-representation in areas where a certain gender is under-represented. Women are under-represented in the salary scale of the advertised position. Therefore, qualified women are encouraged to apply. Moreover, we welcome applications from qualified men. Preference will be given to equally-qualified applicants with disabilities.

Please send your application including a complete curriculum vitae, certificates, and a motivational letter until August 8, 2021 to [email protected]

For more information on the position, please consult the webpage www.uni-hannover.de/en/jobs/4434/ or contact Prof. Matthias Müller, [email protected]

Recrutement

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• Médiathèque

PhD position: Model Predictive Control for Deformable Object Manipulation

Full description in the pdf below: https://seafile.lirmm.fr/f/9d27aca230cb48f28199/

Although many well-established methods exist for handling rigid objects, the manipulation of flexible and soft systems often relies on human intervention. This is primarily due to the complexity associated with the perception and modeling of deformation. The present project will bring together different state-of-the-art methods for modeling, perception, and control of robotic systems to manipulate deformable objects with reliability. The robotic manipulation will be performed by means of Model Predictive Control (MPC).

MPC is an advanced control architecture that computes in real-time the upcoming optimal trajectory satisfying control and state constraints. This is particularly useful since maximal efficiency is often obtained with conditions close to the system constraints. Moreover, thanks to its predictive nature, MPC can anticipate future changes during online computation of the optimal trajectory.

POSITION DESCRIPTION AND RESPONSIBILITIES:

    - Development of MPC schemes for the manipulation of deformable objects;     - Experimental validation with a dual-arm robotic system;     - Modeling through finite element methods (FEM);     - Implementation of perception techniques using depth camera feedback;     - Writing reports and scientific papers in peer-reviewed journals.

ELIGIBILITY:

    - Master's degree on robotics or related topics;     - Solid background on Python and C++ programming;     - Basic knowledge related to numerical optimization;     - B2 spoken and written English;     - Desirable experience with finite element methods;     - Desirable experience with MPC.

WORKING ENVIRONMENT:

The PhD candidate will be supervised by João Cavalcanti Santos and Andrea Cherubini, LIRMM, Montpellier, France. Enjoying over 300 days of sunshine annually, Montpellier offers a delightful Mediterranean climate that fosters a relaxed and enjoyable lifestyle. With its diverse population and international student community, Montpellier offers a cosmopolitan lifestyle enriched by cultural diversity. An internationally renowned laboratory, LIRMM research activities led to relevant results in fields including high-speed industrial robotics, human-robot interaction and construction robotics.

APPLICATION:

Prospective applicants are invited to submit a single PDF file containing both their cover letter and CV to [email protected] and [email protected]. The start work date is flexible, but preferably September/October. The deadline for applications is April 22, 2024. Please ensure that the subject line of the email follows the format "MoPDOM PhD application - [Applicant's Name Surname]".

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PhD Position for Model Predictive Control with Formal Guarantees

Technical university of munich , germany.

04.03.2023,  Wissenschaftliches Personal

The research group Cyber-Physical Systems of Prof. Matthias Althoff at the Technical University of Munich offers a PhD position in the area of model predictive control with formal guarantees. The Technical University of Munich is one of the top research universities in Europe fostering a strong entrepreneurial spirit and international culture.

Expected Starting Date: 01 May 2023-01 October 2023

Closing Date for Applicants: 15 April 2023

Duration: 3 years with a possible extension

Project and Job Description Cyber-physical systems are complex systems that combine physical capabilities with computational capabilities. These include medical devices and systems, process controls, autonomous vehicles, avionic systems, energy systems, robots, manufacturing systems, and smart structures. The increasingly complex  requirements of cyber-physical systems makes it very difficult to control them or even prove that their specification is met. For this reason, model predictive control with formal guarantees is an active area of research. However, there exist no scalable and formally correct synthesis methods for arbitrary nonlinear systems whose algorithmic parameters are automatically tuned.

To address this problem, we propose a synthesis approach that does not rely on any discretization and instead uses optimization techniques in combination with reachability analysis. We plan to translate temporal logic specifications into hybrid automata and compute the product automaton with the system to be controlled. This way, the model predictive control problem can be reformulated as reach-avoid problems for constrained and disturbed hybrid systems. Through optimization, we will obtain an optimal nominal solution and ensure that all other solutions originating from uncertain initial states, disturbances, and sensor noise also meet all constraints.

Among other use cases, we will evaluate our approach on our autonomous vehicle EDGAR. All developed algorithms will be available through our software tools CORA ( cora.in.tum.de ) for reachability analysis and AROC ( aroc.cps.in.tum.de ) for formal controller synthesis.

Previous Work https://mediatum.ub.tum.de/doc/1524264/205786.pdf https://mediatum.ub.tum.de/doc/1454141/411080880765.pdf

Job Specifications • Excellent Master’s degree (or equivalent) in computer science, engineering, or related disciplines (typically mathematics, physics). • Fluency in spoken and written English is required. • Good programming skills in at least one programming language, e.g. MATLAB, C/C++, Python. • Highly motivated and keen on working in an international and interdisciplinary team • Applicants with strong background in the following fields are preferred: ▪ Control Theory ▪ Optimization ▪ Formal methods ▪ Robotics

Context The applicant will be directly advised by Prof. Matthias Althoff ( https://www.ce.cit.tum.de/air/people/prof-dr-ing-matthias-althoff ). Besides excellent skills for conducting innovative science, the candidate should also be talented in implementing research results. The candidate will be integrated in a supportive research environment.

Our Offer PhD remuneration will be in line with the current German collective pay agreement TV-L E13 (around 4500 Euros/month). Technical University of Munich is an equal opportunity employer committed to excellence through diversity. We explicitly encourage women to apply and preference will be given to disabled applicants with equivalent qualifications.

Contact International candidates are highly encouraged to apply. Please submit your complete application (in English or German) via our application form: https://wiki.tum.de/display/cpsforms/Ph.D.+Application . Fill out all mandatory fields (*) and kindly use “Model Predictive Control with Formal Guarantees” as the “Title of Position”. Please do not include a cover letter. Further similar job offerings will be announced on https://www.ce.cit.tum.de/en/air/open-positions/scientific-staff/ .

Data Protection Information: When you apply for a position with the Technical University of Munich (TUM), you are submitting personal information. With regard to personal information, please take note of the Datenschutzhinweise gemäß Art. 13 Datenschutz-Grundverordnung (DSGVO) zur Erhebung und Verarbeitung von personenbezogenen Daten im Rahmen Ihrer Bewerbung. (data protection information on collecting and processing personal data contained in your application in accordance with Art. 13 of the General Data Protection Regulation (GDPR)). By submitting your application, you confirm that you have acknowledged the above data protection information of TUM.

Kontakt: [email protected]

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PhD Position in Model Predictive Control for Flexible Robotic Systems

Catholic university of leuven department mechanical engineering.

Phd Position on Embedded Model Predictive Control for Mechatronic Systems

Model Predictive Control for T-S Fuzzy Markovian Jump Systems Using Dynamic Prediction Optimization

In this paper, the model predictive control (MPC) problem is investigated for the constrained discrete-time Takagi-Sugeno fuzzy Markovian jump systems (FMJSs) under imperfect premise matching rules. To strike a balance between initial feasible region, control performance, and online computation burden, a set of mode-dependent state feedback fuzzy controllers within the frame of dynamic prediction optimizing (DPO)-MPC is delicately designed with the perturbation variables produced by the predictive dynamics. The DPO-MPC controllers are implemented via two stages: at the first stage, terminal constraints sets companied with feedback gain are obtained by solving a “min-max” problem; at the second stage, and a set of perturbations is designed felicitously to enlarge the feasible region. Here, dynamic feedback gains are designed for off-line using matrix factorization technique, while the dynamic controller state is determined for online over a moving horizon to gradually guide the system state from the initial feasible region to the terminal constraint set. Sufficient conditions are provided to rigorously ensure the recursive feasibility of the proposed DPO-MPC scheme and the mean-square stability of the underlying FMJS. Finally, the efficacy of the proposed methods is demonstrated through a robot arm system example.

1 Introduction

With the rapid progress in science and technology, contemporary industrial control systems have grown increasingly intricate, characterized by a proliferation of nonlinearities and uncertainties. The Takagi-Sugeno (T-S) fuzzy modeling approach has emerged as a valuable tool for approximating these complex systems, making T-S fuzzy models integral to the field of control in recent decades. Noteworthy advancements in this area have been documented in various studies [ 23 , 6 , 24 , 12 ] . On the other hand, Markovian jump systems (MJSs) have garnered significant attention from the systems science and engineering community due to their efficacy in describing systems experiencing abrupt changes or random fluctuations. By integrating mode variation with T-S fuzzy rules, fuzzy Markovian jump systems (FMJSs) have captured the interest of researchers across disciplines, particularly in the control domain, being a class of stochastic nonlinear dynamic systems. Previous research efforts in this realm include works such as [ 27 , 29 , 30 ] . It is worth noting that many existing studies predominantly utilize the perfectly matched premises (PMP) framework, also known as parallel distributed compensation (PDC), rather than the imperfectly matched premises (IMP) approach. This preference is primarily attributed to the computational complexity associated with IMP, despite its potential for reduced conservatism, as highlighted in [ 11 ] .

As a cutting-edge modern intelligent control technology, model predictive control (MPC) has demonstrated immense potential in practical applications across various fields [ 8 , 20 , 33 , 19 ] . This is primarily due to its significant advantages in efficiently managing optimization problems involving multiple variables and constraints. Numerous research endeavors have been dedicated to addressing MPC challenges in T-S fuzzy systems [ 4 , 5 , 13 , 7 ] . For instance, in [ 21 ] , the observer-based output feedback MPC problem was explored for a T-S fuzzy system with data loss. Moreover, in [ 22 ] , a Razumikhin approach was introduced for time-delay fuzzy systems, alongside the provision of two robust MPC algorithms. Regrettably, limited results regarding the MPC problem of FMJSs [ 25 ] have been documented in existing literature, primarily due to the challenges associated with ensuring algorithm feasibility in the simultaneous presence of jump modes and fuzzy rules.

From a practical standpoint, computational burden and performance are consistently pivotal issues for MPC strategies, potentially influencing their integration into industrial engineering applications [ 3 ] , particularly when dealing with a large number of fuzzy controller rules within the IMP category. In the case of online MPC strategies, the continual and substantial computation load over a moving horizon, coupled with the strict requirement for the initial system state to belong to the terminal constraint set around the origin, pose significant challenges to MPC’s practicality. On the other hand, off-line MPC strategies excel in reducing online computational complexity but face stability concerns, especially when dealing with model uncertainties and random variations [ 7 , 18 ] . Despite extensive research efforts dedicated to studying both online and off-line MPC strategies, effectively addressing the aforementioned performance issues remains a substantial challenge. Consequently, establishing a comprehensive “off-line to online” design framework for MPC becomes not only essential but also pragmatic to preserve the strengths of both off-line and online approaches while overcoming their limitations.

To this end, an efficient MPC algorithm proposed in [ 9 , 10 ] strikes a balance between off-line and on-line computation, which requires a lower on-line computation than full on-line MPC strategy in [ 18 , 31 ] . Meanwhile, the convex formulation for dynamic prediction optimizing (DPO) on MPC, as discussed in [ 1 , 15 ] , enhances predictive control by aligning controller state dynamics with system state evolution. Building upon this, in [ 16 ] , a new approach was proposed to improve the optimality of DPO-MPC apparently without increasing much online computational load. However, addressing hard constraints in DPO-MPC for complex dynamic systems like FMJSs remains relatively unexplored, motivating the focus of this paper.

In pursuit of enhanced efficiency, a novel MPC algorithm that combines elements of both off-line and online strategies was introduced in [ 9 , 10 ] . This algorithm, requiring less online computation compared to full online MPC strategies [ 18 , 31 ] , offers an additional DoF to extend the initial feasible region. Subsequently, the concept of dynamic prediction optimizing (DPO) within MPC was explored through a convex formulation in works such as [ 1 , 15 ] . This approach empowers the dynamics guiding the predicted controller state to evolve in alignment with the projected system state. Building upon this progress, [ 16 ] introduced a novel method to enhance the optimality of DPO-MPC without significantly increasing online computational overhead. However, despite the practical significance of dynamic systems like FMJSs, research on DPO-MPC problems with stringent constraints is still nascent. The identified knowledge gap forms the primary motivation for our current study.

Our objective in this paper is to present a comprehensive “off-line design and online synthesis” DPO-MPC scheme tailored for a specific discrete-time stochastic FMJSs with constraints. The primary contributions of this paper are detailed as follows. 1)The establishment of an innovative optimizing prediction dynamics framework for MPC design, specifically tailored for T-S FMJSs with hard constraints, marking a pioneering effort in this domain; 2)utilization of mathematical analysis techniques, such as variable substitution, matrix decomposition, and inequality manipulation, to address the non-convexity challenges arising from coupled variables and dynamic state variable introductions; 3)implementation of DPO technology to devise a more adaptable IPM control strategy, effectively mitigating the potential computational overload associated with IPM; 4)development of an “off-line design and online synthesis” scheme, as opposed to the complete online MPC scheme [ 18 , 31 ] , aimed at striking a harmonious balance between computational complexity, initial feasible region and control efficacy. This recursive algorithm structure ensures the requisite mean-square stability of the closing-loop MJSs, thus enhancing overall control performance and system stability.

The remaining sections of the paper are organized as follows. Section 2 presents the formulation of the addressed system model and the DPO-MPC scheme. In Section 3 , the determination of the terminal constraint set and the design of the corresponding control parameters are discussed. The design scheme of perturbation is offered in Section 4 with respect to the “off-line to online synthesis” approach. A single-link robot arm system example is presented in Section 5 and the paper is concluded in Section 6 .

2 Problem Formulation and Preliminaries

2.1 system model.

Let us consider a discrete-time FMJSs using IF-THEN rules: Plant Rule ℏ Planck-constant-over-2-pi \hbar roman_ℏ : IF f ( 1 ) ⁢ ( x s ) superscript 𝑓 1 subscript 𝑥 𝑠 f^{(1)}(x_{s}) italic_f start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) is ℱ ℏ ( 1 ) superscript subscript ℱ Planck-constant-over-2-pi 1 \mathcal{F}_{\hbar}^{(1)} caligraphic_F start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 1 ) end_POSTSUPERSCRIPT , f ( 2 ) ⁢ ( x s ) superscript 𝑓 2 subscript 𝑥 𝑠 f^{(2)}(x_{s}) italic_f start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) is ℱ ℏ ( 2 ) superscript subscript ℱ Planck-constant-over-2-pi 2 \mathcal{F}_{\hbar}^{(2)} caligraphic_F start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( 2 ) end_POSTSUPERSCRIPT , … … \ldots … , and f ( r ) ⁢ ( x s ) superscript 𝑓 𝑟 subscript 𝑥 𝑠 f^{(r)}(x_{s}) italic_f start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) is ℱ ℏ ( r ) superscript subscript ℱ Planck-constant-over-2-pi 𝑟 \mathcal{F}_{\hbar}^{(r)} caligraphic_F start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_r ) end_POSTSUPERSCRIPT ,

The stochastic process ζ s subscript 𝜁 𝑠 \zeta_{s} italic_ζ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT denotes a homogeneous Markov chain in a finite state space ℐ N subscript ℐ 𝑁 \mathcal{I}_{N} caligraphic_I start_POSTSUBSCRIPT italic_N end_POSTSUBSCRIPT with the transition probability

Aided by the T-S fuzzy approach, we establish the overall FMJSs with ζ s = ı subscript 𝜁 𝑠 italic-ı \zeta_{s}=\imath italic_ζ start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT = italic_ı as follows:

with ℱ ℏ ( α ) ⁢ ( f ( α ) ⁢ ( x s ) ) superscript subscript ℱ Planck-constant-over-2-pi 𝛼 superscript 𝑓 𝛼 subscript 𝑥 𝑠 \mathcal{F}_{\hbar}^{(\alpha)}(f^{(\alpha)}(x_{s})) caligraphic_F start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT ( italic_f start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ) referring to the grade of membership of f ( α ) ⁢ ( x s ) superscript 𝑓 𝛼 subscript 𝑥 𝑠 f^{(\alpha)}(x_{s}) italic_f start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) in ℱ ℏ ( α ) superscript subscript ℱ Planck-constant-over-2-pi 𝛼 \mathcal{F}_{\hbar}^{(\alpha)} caligraphic_F start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT start_POSTSUPERSCRIPT ( italic_α ) end_POSTSUPERSCRIPT . And θ ℏ ⁢ ( f ⁢ ( x s ) ) subscript 𝜃 Planck-constant-over-2-pi 𝑓 subscript 𝑥 𝑠 \theta_{\hbar}(f(x_{s})) italic_θ start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT ( italic_f ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ) implies the standard membership function of rule ℏ Planck-constant-over-2-pi \hbar roman_ℏ . For ∀ ℏ ∈ ℐ T for-all Planck-constant-over-2-pi subscript ℐ 𝑇 \forall\hbar\in\mathcal{I}_{T} ∀ roman_ℏ ∈ caligraphic_I start_POSTSUBSCRIPT italic_T end_POSTSUBSCRIPT , we have θ ℏ ⁢ ( f ⁢ ( x s ) ) ≥ 0 subscript 𝜃 Planck-constant-over-2-pi 𝑓 subscript 𝑥 𝑠 0 \theta_{\hbar}(f(x_{s}))\geq 0 italic_θ start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT ( italic_f ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ) ≥ 0 and ∑ ℏ = 1 t θ ℏ ⁢ ( f ⁢ ( x s ) ) = 1 superscript subscript Planck-constant-over-2-pi 1 𝑡 subscript 𝜃 Planck-constant-over-2-pi 𝑓 subscript 𝑥 𝑠 1 \sum_{\hbar=1}^{t}\theta_{\hbar}(f(x_{s}))=1 ∑ start_POSTSUBSCRIPT roman_ℏ = 1 end_POSTSUBSCRIPT start_POSTSUPERSCRIPT italic_t end_POSTSUPERSCRIPT italic_θ start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT ( italic_f ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ) = 1 . For notational brevity, θ ℏ subscript 𝜃 Planck-constant-over-2-pi \theta_{\hbar} italic_θ start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT is denoted as θ ℏ ⁢ ( f ⁢ ( x s ) ) subscript 𝜃 Planck-constant-over-2-pi 𝑓 subscript 𝑥 𝑠 \theta_{\hbar}(f(x_{s})) italic_θ start_POSTSUBSCRIPT roman_ℏ end_POSTSUBSCRIPT ( italic_f ( italic_x start_POSTSUBSCRIPT italic_s end_POSTSUBSCRIPT ) ) for the subsequent analysis.

The FMJSs will be affected by the constraints imposed by engineering practices on inputs and states as follows: