Advances in Analysis and Control of Systems with Uncertainties II

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 June 2024 | Viewed by 6919

Special Issue Editor


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Guest Editor
The Galilee Research Center for Applied Mathematics, Braude College of Engineering, Karmiel, Israel
Interests: asymptotic methods; differential games; generalized functions; hybrid systems; optimal control; robust control; singular optimal control problems and singular differential games; singularly perturbed problems; stochastic difference and differential equations; systems theory; time delay systems
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Special Issue Information

Dear Colleagues,

This issue is a continuation of our previous Special Issue on "Systems with Uncertainties". We will provide an opportunity to present recent developments in theory and various real-life applications of systems with uncertainties, their analysis and designs for the robust control of such systems. This Special Issue will address the following, non-exhaustive list of topics:

qualitative analysis of uncertain systems, such as the stability, stabilizability, detectability, stabilization, controllability, observability and reachability of such systems; robust control of uncertain systems, including sliding mode control, H∞ control, low-chattering and chattering-free control, and game-based control; filtering and/or estimation of uncertain systems; real-life uncertain systems, including qualitative analysis and/or robust control design.

It should be noted that the Special Issue is open to receiving further ideas, in addition to the aforementioned topics.

We hope that this initiative will be attractive to experts in the theory of systems with uncertainties and its various real-life applications. We encourage you to submit your current research for inclusion in the Special Issue.

Prof. Dr. Valery Y. Glizer
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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • systems with euclidean space bounded uncertainties
  • systems with polytopic uncertainties
  • systems with square integrable uncertainties
  • systems with matched uncertainties
  • stability
  • stabilizability
  • detectability
  • stabilization
  • controllability
  • observability
  • reachability
  • sliding mode control of uncertain systems
  • h∞ control of uncertain systems
  • low-chattering control of uncertain systems
  • chattering-free control of uncertain systems
  • game-based control of uncertain systems
  • filtering of uncertain systems
  • estimation of uncertain systems
  • real-life uncertain systems

Published Papers (8 papers)

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Research

42 pages, 583 KiB  
Article
Robust Solution of the Multi-Model Singular Linear-Quadratic Optimal Control Problem: Regularization Approach
by Valery Y. Glizer
Axioms 2023, 12(10), 955; https://doi.org/10.3390/axioms12100955 - 10 Oct 2023
Viewed by 954
Abstract
We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve [...] Read more.
We consider a finite horizon multi-model linear-quadratic optimal control problem. For this problem, we treat the case where the problem’s functional does not contain a control function. The latter means that the problem under consideration is a singular optimal control problem. To solve this problem, we associate it with a new optimal control problem for the same multi-model system. The functional in this new problem is the sum of the original functional and an integral of the square of the Euclidean norm of the vector-valued control with a small positive weighting coefficient. Thus, the new problem is regular. Moreover, it is a multi-model cheap control problem. Using the solvability conditions (Robust Maximum Principle), the solution of this cheap control problem is reduced to the solution of the following three problems: (i) a terminal-value problem for an extended matrix Riccati type differential equation; (ii) an initial-value problem for an extended vector linear differential equation; (iii) a nonlinear optimization (mathematical programming) problem. We analyze an asymptotic behavior of these problems. Using this asymptotic analysis, we design the minimizing sequence of state-feedback controls for the original multi-model singular optimal control problem, and obtain the infimum of the functional of this problem. We illustrate the theoretical results with an academic example. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
14 pages, 330 KiB  
Article
Non-Zero Sum Nash Game for Discrete-Time Infinite Markov Jump Stochastic Systems with Applications
by Yueying Liu, Zhen Wang and Xiangyun Lin
Axioms 2023, 12(9), 882; https://doi.org/10.3390/axioms12090882 - 15 Sep 2023
Cited by 2 | Viewed by 649
Abstract
This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a [...] Read more.
This paper is to study finite horizon linear quadratic (LQ) non-zero sum Nash game for discrete-time infinite Markov jump stochastic systems (IMJSSs). Based on the theory of stochastic analysis, a countably infinite set of coupled generalized algebraic Riccati equations are solved and a necessary and sufficient condition for the existence of Nash equilibrium points is obtained. From a new perspective, the finite horizon mixed robust H2/H control is investigated, and summarize the relationship between Nash game and H2/H control problem. Moreover, the feasibility and validity of the proposed method has been proved by applying it to a numerical example. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
21 pages, 515 KiB  
Article
The Stability Analysis of Linear Systems with Cauchy—Polynomial-Vandermonde Matrices
by Mutti-Ur Rehman, Jehad Alzabut, Nahid Fatima and Tulkin H. Rasulov
Axioms 2023, 12(9), 831; https://doi.org/10.3390/axioms12090831 - 28 Aug 2023
Viewed by 666
Abstract
The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new [...] Read more.
The numerical approximation of both eigenvalues and singular values corresponding to a class of totally positive Bernstein–Vandermonde matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices are well studied and investigated in the literature. We aim to present some new results for the numerical approximation of the largest singular values corresponding to Bernstein–Vandermonde, Bernstein–Bezoutian, Cauchy—polynomial-Vandermonde and quasi-rational Bernstein–Vandermonde structured matrices. The numerical approximation for the reciprocal of the largest singular values returns the structured singular values. The new results for the numerical approximation of bounds from below for structured singular values are accomplished by computing the largest singular values of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and quasi-rational Bernstein–Vandermonde structured matrices. Furthermore, we present the spectral properties of totally positive Bernstein–Vandermonde structured matrices, Bernstein–Bezoutian structured matrices, Cauchy—polynomial-Vandermonde structured matrices, and structured quasi-rational Bernstein–Vandermonde matrices by computing the eigenvalues, singular values, structured singular values and its lower and upper bounds and condition numbers. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
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25 pages, 11822 KiB  
Article
Model-Free Sliding Mode Enhanced Proportional, Integral, and Derivative (SMPID) Control
by Quanmin Zhu
Axioms 2023, 12(8), 721; https://doi.org/10.3390/axioms12080721 - 25 Jul 2023
Cited by 3 | Viewed by 999
Abstract
This study proposes a type of Sliding Mode-based Proportional, Integral, and Derivative (SMPID) controllers to establish a model-free (treat dynamic plants as a whole uncertainty) sliding model control (MFSMC) platform for Bounded-Input and Bounded-Output (BIBO) dynamic systems. The SMPID design (1) proposes a [...] Read more.
This study proposes a type of Sliding Mode-based Proportional, Integral, and Derivative (SMPID) controllers to establish a model-free (treat dynamic plants as a whole uncertainty) sliding model control (MFSMC) platform for Bounded-Input and Bounded-Output (BIBO) dynamic systems. The SMPID design (1) proposes a sliding mode error (rather than error) as the PID input, (2) directly links to Lyapunov asymptotic stability to provide total robust nonlinear dynamic inversion (NDI), and (3) reduces the chattering effects in terms of Lyapunov definite positive stability. Further, the study proposes a general SMC framework to accommodate asymptotic time stabilisation and finite-time stabilisation for both model-based and model-free designs. A U-control framework is presented to integrate the SMPID control (for NDI) and an invariant control (IC) (for specifying the whole control system’s dynamic and static responses), which significantly relaxes the PID tunings and generates the specified performance. To provide assurance and guidance for applications and expansions, this study presents the relevant fundamental analyses and transparent simulated bench tests. It should be noted that the new SMPID in forms of u=SMPID(σ(e))=PID(sliding-mode) is different from that studied u=sliding-mode(PID(e)) in expression and functionality. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
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13 pages, 455 KiB  
Article
Robust Stability of Switched-Interval Positive Linear Systems with All Modes Unstable Using the Φ-Dependent Dwell Time Technique
by Qiang Yu and Xiujuan Jiang
Axioms 2023, 12(7), 686; https://doi.org/10.3390/axioms12070686 - 13 Jul 2023
Viewed by 697
Abstract
In this study, some stability and robust stability conditions for switched positive linear systems in which all subsystems are unstable in continuous time and discrete time were obtained using the Φ-dependent dwell time technique and the discretized co-positive Lyapunov functions approach, respectively. The [...] Read more.
In this study, some stability and robust stability conditions for switched positive linear systems in which all subsystems are unstable in continuous time and discrete time were obtained using the Φ-dependent dwell time technique and the discretized co-positive Lyapunov functions approach, respectively. The co-positive Lyapunov functions constructed in this study are functions of time during the dwell time, and after that, they are independent of time. In addition, the above method was applied to switched-interval positive systems, and corresponding conclusions are presented. The Φ-dependent dwell time technique used in this paper is more effective than the dwell time and mode-dependent dwell time used in other studies. The results are verified with an illustrative example. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
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12 pages, 275 KiB  
Article
A New Reverse Extended Hardy–Hilbert’s Inequality with Two Partial Sums and Parameters
by Jianquan Liao and Bicheng Yang
Axioms 2023, 12(7), 678; https://doi.org/10.3390/axioms12070678 - 10 Jul 2023
Cited by 1 | Viewed by 549
Abstract
By using the methods of real analysis and the mid-value theorem, we introduce some lemmas and obtain a new reverse extended Hardy–Hilbert’s inequality with two partial sums and multi-parameters. We also give a few equivalent conditions of the best possible constant factor related [...] Read more.
By using the methods of real analysis and the mid-value theorem, we introduce some lemmas and obtain a new reverse extended Hardy–Hilbert’s inequality with two partial sums and multi-parameters. We also give a few equivalent conditions of the best possible constant factor related to several parameters in the new inequality. Some particular inequalities are deduced. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
33 pages, 3836 KiB  
Article
Forecasting Water Consumption in the Yangtze River Delta Based on Deformable Cumulative Multivariable Grey Model
by Zhengran Qiao and Wei Yang
Axioms 2023, 12(7), 655; https://doi.org/10.3390/axioms12070655 - 01 Jul 2023
Viewed by 707
Abstract
The intensified contradiction between water resources and social development has restricted the development of the Yangtze River Delta. Due to the importance of water consumption in relieving this contradiction, this paper proposes a novel cumulative multivariable grey model with a high performance to [...] Read more.
The intensified contradiction between water resources and social development has restricted the development of the Yangtze River Delta. Due to the importance of water consumption in relieving this contradiction, this paper proposes a novel cumulative multivariable grey model with a high performance to predict the water consumption. Firstly, the grey correlation analysis is applied to study the influencing factors, and then the DGM(1,N) with deformable accumulation (DDGM(1,N) model) is constructed and used to predict the water consumption. The results show that the resident population has a significant impact on the water consumption, and the performance of the DDGM(1,N) model is better than the other two grey models. Secondly, the proposed novel grey model is applied to predict the water consumption in 17 cities in the Yangtze River Delta, and the predicted water consumption in Zhejiang and Shanghai indicates a downward trend, while the predicated water consumption in some cities of the Anhui Province presents an upward trend, such as Chizhou, Chuzhou, Wuhu and Tongling. Finally, some policy implications are provided that correspond to the population growth and three major industries in different situations. This paper enriches the research method and prediction analysis used for the water consumption, and the findings can provide some decision-making references for water resources management. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
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14 pages, 1963 KiB  
Article
Robust Consensus in a Class of Fractional-Order Multi-Agent Systems with Interval Uncertainties Using the Existence Condition of Hermitian Matrices
by Mohammadreza Riazat, Aydin Azizi, Mojtaba Naderi Soorki and Abbasali Koochakzadeh
Axioms 2023, 12(1), 65; https://doi.org/10.3390/axioms12010065 - 07 Jan 2023
Cited by 3 | Viewed by 1166
Abstract
This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are [...] Read more.
This study outlines the necessary and sufficient criteria for swarm stability asymptotically, meaning consensus in a class of fractional-order multi-agent systems (FOMAS) with interval uncertainties for both fractional orders 0 < α < 1 and 1 < α < 2. The constraints are determined by the graph topology, agent dynamics, and neighbor interactions. It is demonstrated that the fractional-order interval multi-agent system achieves consensus if and only if there are some Hermitian matrices that satisfy a particular kind of complex Lyapunov inequality for all of the system vertex matrices. This is done by using the existence condition of the Hermitian matrices in a Lyapunov inequality. To do this, at first it is shown under which conditions a multi-agent system with unstable agents can still achieve consensus. Then, using a lemma and a theory, the Lyapunov inequality regarding the negativity of the maximum eigenvalue of an augmented matrix of a FOMAS is used to find some Hermitian matrices by checking only a limited number of system vertex matrices. As a result, the necessary and sufficient conditions to reach consensus in a FOMAS in the presence of internal uncertainties are obtained according to the Lyapunov inequalities. Using the main theory of the current paper, instead of countless matrices, only a limited number of vertex matrices need to be used in Lyapunov inequalities to find some Hermitian matrices. As a confirmation of the notion, some instances from numerical simulation are also provided at the end of the paper. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties II)
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