Pub Date : 2020-01-01DOI: 10.56082/annalsarscimath.2020.1-2.538
Kliti Kodra, N. Zhong, Z. Gajic
In this paper, we point out important observations on time-scale decomposition of linear singularly perturbed systems. It has been established in the control literature that the asymptotically stable fast modes of a singularly perturbed system decay rapidly in a boundary layer interval when the perturbation parameter is very small hence the slow subsystem can serve as a good approximation of the original model. We observe that while this is the case in the steady state, it is not true during the transient response for a strictly proper system with highly damped and highly oscillatory modes. Instead, the fast subsystem provides a very good approximation of the original model’s response but with a DC gain offset. We propose a correction to rectify the DC gain offset and illustrate the findings using an islanded microgrid electric power system model.
{"title":"REMARKS ON TIME-SCALE DECOMPOSITION USING SINGULAR PERTURBATIONS WITH APPLICATIONS","authors":"Kliti Kodra, N. Zhong, Z. Gajic","doi":"10.56082/annalsarscimath.2020.1-2.538","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.538","url":null,"abstract":"In this paper, we point out important observations on time-scale decomposition of linear singularly perturbed systems. It has been established in the control literature that the asymptotically stable fast modes of a singularly perturbed system decay rapidly in a boundary layer interval when the perturbation parameter is very small hence the slow subsystem can serve as a good approximation of the original model. We observe that while this is the case in the steady state, it is not true during the transient response for a strictly proper system with highly damped and highly oscillatory modes. Instead, the fast subsystem provides a very good approximation of the original model’s response but with a DC gain offset. We propose a correction to rectify the DC gain offset and illustrate the findings using an islanded microgrid electric power system model.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87241261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.56082/annalsarscimath.2020.1-2.581
A. Doban, M. Lazar
In this paper we propose an input-to-state stability (ISS) criterion for continuous–time systems based on a finite–time decrease condition for a positive definite function of the norm of the state. This yields a so–called ISS finite–time Lyapunov function, which allows for easier choice of candidate functions compared to standard ISS Lyapunov functions. An alternative converse ISS theorem in terms of ISS finite– time Lyapunov functions is also provided. Moreover, we prove that ISS finite–time Lyapunov functions are equivalent with standard ISS Lyapunov functions using a Massera–type construction. The developed ISS framework can be utilized in combination with Sontag’s “universal” stabilisation formula to develop input–to–state stabilizing control laws for continuous–time nonlinear systems that are affine in the control and disturbance inputs, respectively. MSC: 93C10, 93D09, 93D30, 93D15
{"title":"INPUT-TO-STATE STABILITY FINITE-TIME LYAPUNOV FUNCTIONS FOR CONTINUOUS-TIME SYSTEMS","authors":"A. Doban, M. Lazar","doi":"10.56082/annalsarscimath.2020.1-2.581","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.581","url":null,"abstract":"In this paper we propose an input-to-state stability (ISS) criterion for continuous–time systems based on a finite–time decrease condition for a positive definite function of the norm of the state. This yields a so–called ISS finite–time Lyapunov function, which allows for easier choice of candidate functions compared to standard ISS Lyapunov functions. An alternative converse ISS theorem in terms of ISS finite– time Lyapunov functions is also provided. Moreover, we prove that ISS finite–time Lyapunov functions are equivalent with standard ISS Lyapunov functions using a Massera–type construction. The developed ISS framework can be utilized in combination with Sontag’s “universal” stabilisation formula to develop input–to–state stabilizing control laws for continuous–time nonlinear systems that are affine in the control and disturbance inputs, respectively. MSC: 93C10, 93D09, 93D30, 93D15","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87269258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.56082/annalsarscimath.2020.1-2.99
N. Das, Swarupa Roy
In this paper we have shown that if S ∈ L(L 2 a (dAα)) and Θ (α) S (x, y)Θ(α) T (x, y)(K(α) (x, y))2 ≈ Θ (α) ST (x, y)(K(α) (x, y))2 for all x, y ∈ D and for all T ∈ L(L 2 a (dAα)), then S = T (α) φ for some φ ∈ H∞(D) and the matrix of S is lower triangular, where Θ(α) S (x, y) for S ∈ L(L 2 a (dAα)) is a function on D × D meromorphic in x and conjugate meromorphic in y. Further, we show that if ψ, φ ∈ L∞(D), R(α) ∈ L(L 2 a (dAα)), then Θ(α) T (α) φ (x, y)Θ(α) S (α) ψ (x, y)(K(α) (x, y))2 ≈ Θ (α) R(α) (x, y) ·(K(α) (x, y))2 holds for all x, y ∈ D if and only if there exists β ∈ C such that φ ≡ β and R(α) = S (α) βψ .
{"title":"TOEPLITZ AND HANKEL OPERATORS ON WEIGHTED BERGMAN SPACES","authors":"N. Das, Swarupa Roy","doi":"10.56082/annalsarscimath.2020.1-2.99","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.99","url":null,"abstract":"In this paper we have shown that if S ∈ L(L 2 a (dAα)) and Θ (α) S (x, y)Θ(α) T (x, y)(K(α) (x, y))2 ≈ Θ (α) ST (x, y)(K(α) (x, y))2 for all x, y ∈ D and for all T ∈ L(L 2 a (dAα)), then S = T (α) φ for some φ ∈ H∞(D) and the matrix of S is lower triangular, where Θ(α) S (x, y) for S ∈ L(L 2 a (dAα)) is a function on D × D meromorphic in x and conjugate meromorphic in y. Further, we show that if ψ, φ ∈ L∞(D), R(α) ∈ L(L 2 a (dAα)), then Θ(α) T (α) φ (x, y)Θ(α) S (α) ψ (x, y)(K(α) (x, y))2 ≈ Θ (α) R(α) (x, y) ·(K(α) (x, y))2 holds for all x, y ∈ D if and only if there exists β ∈ C such that φ ≡ β and R(α) = S (α) βψ .","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87581824","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-01-01DOI: 10.56082/annalsarscimath.2020.1-2.329
Rovana Boruga, M. Megan
The paper considers a general concept of polynomial dichotomy which includes as particular cases some well-known dichotomy concepts. The main objective is to obtain some characterizations of the nonuniform polynomial dichotomy behavior with respect to a family of norms compatible with the projection families
{"title":"NONUNIFORM POLYNOMIAL DICHOTOMY WITH LYAPUNOV TYPE NORMS","authors":"Rovana Boruga, M. Megan","doi":"10.56082/annalsarscimath.2020.1-2.329","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.329","url":null,"abstract":"The paper considers a general concept of polynomial dichotomy which includes as particular cases some well-known dichotomy concepts. The main objective is to obtain some characterizations of the nonuniform polynomial dichotomy behavior with respect to a family of norms compatible with the projection families","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83959834","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2018-09-18DOI: 10.56082/annalsarscimath.2020.1-2.459
V. Crai, M. Megan
The main objective of this paper is to give a characterization in terms of Lyapunov functions for trichotomy with growth rates of evolution operators in Banach spaces.
本文的主要目的是给出Banach空间中具有演化算子增长率的三切分的Lyapunov函数的刻划。
{"title":"LYAPUNOV FUNCTIONS FOR TRICHOTOMY WITH GROWTH RATES OF EVOLUTION OPERATORS IN BANACH SPACES","authors":"V. Crai, M. Megan","doi":"10.56082/annalsarscimath.2020.1-2.459","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.459","url":null,"abstract":"The main objective of this paper is to give a characterization in terms of Lyapunov functions for trichotomy with growth rates of evolution operators in Banach spaces.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-09-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84179155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.56082/annalsarscimath.2020.1-2.226
I. Ivanov
We investigate a set of nonlinear matrix equations with nonnegative matrix coefficients which has arisen in applied sciences. There are papers where the minimal nonnegative solution of the set of nonlinear matrix equations is computed applying the different procedures. The alternate linear implicit method and its modifications have intensively investigated because they have simple computational scheme. We construct a new decoupled modification of the alternate linear implicit procedure to compute the minimal nonnegative solution of the considered set of equations. The convergence properties of the proposed iteration are derived and a sufficient condition for convergence is derived. The performance of the proposed algorithm is illustrated on several numerical examples. On the basis of the experiments we derive conclusions for applicability of the computational schemes.
{"title":"ITERATIVE COMPUTING THE MINIMAL SOLUTION OF THE COUPLED NONLINEAR MATRIX EQUATIONS IN TERMS OF NONNEGATIVE MATRICES","authors":"I. Ivanov","doi":"10.56082/annalsarscimath.2020.1-2.226","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.226","url":null,"abstract":"We investigate a set of nonlinear matrix equations with nonnegative matrix coefficients which has arisen in applied sciences. There are papers where the minimal nonnegative solution of the set of nonlinear matrix equations is computed applying the different procedures. The alternate linear implicit method and its modifications have intensively investigated because they have simple computational scheme. We construct a new decoupled modification of the alternate linear implicit procedure to compute the minimal nonnegative solution of the considered set of equations. The convergence properties of the proposed iteration are derived and a sufficient condition for convergence is derived. The performance of the proposed algorithm is illustrated on several numerical examples. On the basis of the experiments we derive conclusions for applicability of the computational schemes.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90868405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 1900-01-01DOI: 10.56082/annalsarscimath.2020.1-2.364
V. Prepelita, Tiberiu Vasilache
The controllability of a class of 2D linear time varying continuous time control systems is studied. The state space representation is provided and the formulas of the states and the input-output map of these systems are derived. The fundamental concepts of controllability and reachability are analysed and suitable controllability and reachability Gramians are constructed to characterize the controllable and the reachable time varying systems. In the case of time invariant 2D systems, some algorithms are developed to calculate different controllability Gramians as solutions of adequate Lyapunov type equations. Corresponding Matlab programs are implemented to solve these Lyapunov equations.
{"title":"CONTROLLABILITY AND GRAMIANS OF 2D CONTINUOUS TIME LINEAR SYSTEMS","authors":"V. Prepelita, Tiberiu Vasilache","doi":"10.56082/annalsarscimath.2020.1-2.364","DOIUrl":"https://doi.org/10.56082/annalsarscimath.2020.1-2.364","url":null,"abstract":"The controllability of a class of 2D linear time varying continuous time control systems is studied. The state space representation is provided and the formulas of the states and the input-output map of these systems are derived. The fundamental concepts of controllability and reachability are analysed and suitable controllability and reachability Gramians are constructed to characterize the controllable and the reachable time varying systems. In the case of time invariant 2D systems, some algorithms are developed to calculate different controllability Gramians as solutions of adequate Lyapunov type equations. Corresponding Matlab programs are implemented to solve these Lyapunov equations.","PeriodicalId":38807,"journal":{"name":"Annals of the Academy of Romanian Scientists: Series on Mathematics and its Applications","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"1900-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88086423","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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