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Axioms
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Advances in Analysis and Control of Systems with Uncertainties
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Advances in Analysis and Control of Systems with Uncertainties

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

Deadline for manuscript submissions: closed (30 April 2021) | Viewed by 13689

Special Issue Editor

Prof. Dr. Valery Y. Glizer

E-Mail Website
Guest Editor
The Galilee Research Center for Applied Mathematics, Braude College of Engineering, Karmiel 2161002, 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
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

It is our pleasure to announce the launch of a new Special Issue of Axioms. This Special Issue is devoted to "Analysis and Control of Uncertain Continuous/Discrete-Time Undelayed/Delayed Systems".

With this Special Issue, we will provide an opportunity to present recent developments in theory and various real-life applications of analysis of systems with uncertainties (deterministic or stochastic, parametric or non-parametric, matched or unmatched) and design of robust control for such systems. This Special Issue will address the following non-exhaustive list of topics: qualitative analysis of uncertain systems, such as stability, stabilizability, detectability, controllability, and observability 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; etc.

It should be noted that the Special Issue is open to receiving further ideas, apart from 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 to be included 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.

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Published Papers (7 papers)

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Research

15 pages, 343 KiB  
Article
Robust Stability of Time-Varying Markov Jump Linear Systems with Respect to a Class of Structured, Stochastic, Nonlinear Parametric Uncertainties
by Vasile Dragan and Samir Aberkane
Axioms 2021, 10(3), 148; https://doi.org/10.3390/axioms10030148 - 5 Jul 2021
Cited by 7 | Viewed by 1783
Abstract
This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise [...] Read more.
This note is devoted to a robust stability analysis, as well as to the problem of the robust stabilization of a class of continuous-time Markovian jump linear systems subject to block-diagonal stochastic parameter perturbations. The considered parametric uncertainties are of multiplicative white noise type with unknown intensity. In order to effectively address the multi-perturbations case, we use scaling techniques. These techniques allow us to obtain an estimation of the lower bound of the stability radius. A first characterization of a lower bound of the stability radius is obtained in terms of the unique bounded and positive semidefinite solutions of adequately defined parameterized backward Lyapunov differential equations. A second characterization is given in terms of the existence of positive solutions of adequately defined parameterized backward Lyapunov differential inequalities. This second result is then exploited in order to solve a robust control synthesis problem. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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16 pages, 1091 KiB  
Article
Two Inverse Problems Solution by Feedback Tracking Control
by Vladimir Turetsky
Axioms 2021, 10(3), 137; https://doi.org/10.3390/axioms10030137 - 28 Jun 2021
Cited by 1 | Viewed by 1794
Abstract
Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing [...] Read more.
Two inverse ill-posed problems are considered. The first problem is an input restoration of a linear system. The second one is a restoration of time-dependent coefficients of a linear ordinary differential equation. Both problems are reformulated as auxiliary optimal control problems with regularizing cost functional. For the coefficients restoration problem, two control models are proposed. In the first model, the control coefficients are approximated by the output and the estimates of its derivatives. This model yields an approximating linear-quadratic optimal control problem having a known explicit solution. The derivatives are also obtained as auxiliary linear-quadratic tracking controls. The second control model is accurate and leads to a bilinear-quadratic optimal control problem. The latter is tackled in two ways: by an iterative procedure and by a feedback linearization. Simulation results show that a bilinear model provides more accurate coefficients estimates. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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31 pages, 458 KiB  
Article
Nash Equilibrium Sequence in a Singular Two-Person Linear-Quadratic Differential Game
by Valery Y. Glizer
Axioms 2021, 10(3), 132; https://doi.org/10.3390/axioms10030132 - 25 Jun 2021
Cited by 4 | Viewed by 1676
Abstract
A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are [...] Read more.
A finite-horizon two-person non-zero-sum differential game is considered. The dynamics of the game is linear. Each of the players has a quadratic functional on its own disposal, which should be minimized. The case where weight matrices in control costs of one player are singular in both functionals is studied. Hence, the game under the consideration is singular. A novel definition of the Nash equilibrium in this game (a Nash equilibrium sequence) is proposed. The game is solved by application of the regularization method. This method yields a new differential game, which is a regular Nash equilibrium game. Moreover, the new game is a partial cheap control game. An asymptotic analysis of this game is carried out. Based on this analysis, the Nash equilibrium sequence of the pairs of the players’ state-feedback controls in the singular game is constructed. The expressions for the optimal values of the functionals in the singular game are obtained. Illustrative examples are presented. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
17 pages, 641 KiB  
Article
Analyzing Uncertain Dynamical Systems after State-Space Transformations into Cooperative Form: Verification of Control and Fault Diagnosis
by Julia Kersten, Andreas Rauh and Harald Aschemann
Axioms 2021, 10(2), 88; https://doi.org/10.3390/axioms10020088 - 10 May 2021
Cited by 1 | Viewed by 2097
Abstract
When modeling real-life applications, uncertainty appears in the form of, for example, modeling approximations, measurement errors, or simply physical restrictions. Those uncertainties can either be treated as stochastic or as bounded, with known limits in the form of intervals. The latter is considered [...] Read more.
When modeling real-life applications, uncertainty appears in the form of, for example, modeling approximations, measurement errors, or simply physical restrictions. Those uncertainties can either be treated as stochastic or as bounded, with known limits in the form of intervals. The latter is considered in this paper for a real-life application in the form of an electrical circuit. This is reasonable because the electrical circuit is subject to uncertainties, mainly due to circuit element tolerances and variable load conditions. Since conservative worst-case limits for such parameters are commonly known, interval methods can be applied. The aim of this paper is to demonstrate a possible overall handling of the given uncertain system. Firstly, this includes a control and a reliable computation of the states’ interval enclosures. On the one hand, this can be used to predict the system’s behavior, and on the other hand to verify the control numerically. Here, the implemented feedback control is based on linear matrix inequalities (LMIs) and the states are predicted using an interval enclosure technique based on cooperativity. Since the original system is not cooperative, a transformation is performed. Finally, an observer is implemented as a diagnosis tool regarding faulty measurements or component failures. Since adding a state-of-the-art observer would destroy this structure, a cooperativity-preserving method is applied. Overall, this paper combines methods from robust control design and interval-based evaluations, and presents a suitable observer technique to show the applicability of the presented methods. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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13 pages, 623 KiB  
Article
An Analytic and Numerical Investigation of a Differential Game
by Aviv Gibali and Oleg Kelis
Axioms 2021, 10(2), 66; https://doi.org/10.3390/axioms10020066 - 17 Apr 2021
Cited by 2 | Viewed by 1785
Abstract
In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot [...] Read more.
In this paper we present an appropriate singular, zero-sum, linear-quadratic differential game. One of the main features of this game is that the weight matrix of the minimizer’s control cost in the cost functional is singular. Due to this singularity, the game cannot be solved either by applying the Isaacs MinMax principle, or the Bellman–Isaacs equation approach. As an application, we introduced an interception differential game with an appropriate regularized cost functional and developed an appropriate dual representation. By developing the variational derivatives of this regularized cost functional, we apply Popov’s approximation method and show how the numerical results coincide with the dual representation. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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15 pages, 811 KiB  
Article
Nonlinear Control System Design of an Underactuated Robot Based on Operator Theory and Isomorphism Scheme
by Mingcong Deng and Shotaro Kubota
Axioms 2021, 10(2), 62; https://doi.org/10.3390/axioms10020062 - 16 Apr 2021
Cited by 3 | Viewed by 2000
Abstract
The number of actuators of an underactuated robot is less than its degree of freedom. In other words, underactuated robots can be designed with fewer actuators than fully actuated ones. Although an underactuated robot is more complex than a fully actuated robot, it [...] Read more.
The number of actuators of an underactuated robot is less than its degree of freedom. In other words, underactuated robots can be designed with fewer actuators than fully actuated ones. Although an underactuated robot is more complex than a fully actuated robot, it has many advantages, such as energy, material, and space saving. Therefore, it has high research value in both control theory and practical applications. Swing-up is a mechanism with two links, which mimics a gymnast performing a horizontal bar movement. Over the past few decades, many sufficiently robust control techniques have been developed for a fully actuated robot but almost none of them can be directly applicable to an underactuated robot system. The reason is that such control techniques require certain assumptions that are valid only for fully actuated robot systems but not for underactuated ones. In this paper, a control system design method for underactuated robots based on operator theory and an isomorphism scheme is first proposed. Bezout identity is designed using isomorphism. The effectiveness of the design method is confirmed by simulation. The simulation results show that the performances, such as robust stability and response time, of an underactuated robot control system are improved. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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8 pages, 217 KiB  
Article
A Weak Turnpike Property for Perturbed Dynamical Systems with a Lyapunov Function
by Alexander J. Zaslavski
Axioms 2021, 10(2), 45; https://doi.org/10.3390/axioms10020045 - 29 Mar 2021
Cited by 3 | Viewed by 1439
Abstract
In this work, we obtain a weak version of the turnpike property of trajectories of perturbed discrete disperse dynamical systems, which have a prototype in mathematical economics. Full article
(This article belongs to the Special Issue Advances in Analysis and Control of Systems with Uncertainties)
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