Research

42 pages, 583 KiB

Open AccessArticle

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 958

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

Open AccessArticle

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 654

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

H_{2} / H_{\infty}

control is investigated, and summarize the relationship between Nash game and

H_{2} / H_{\infty}

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

Open AccessArticle

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 674

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

Open AccessArticle

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 1006

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 = S M P I D (σ (e)) = P I D (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

Open AccessArticle

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 706

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

Open AccessArticle

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 557

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

Open AccessArticle

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 715

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

Open AccessArticle

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 1174

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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