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EN
In the paper a Lyapunov matrices approach to the parametric optimization problem of a time-delay system with two commensurate delays and a PI-controller is presented. The value of integral quadratic performance index is equal to the value of the Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix. In the paper is presented the example of a scalar system with two delays and a PI controller.
EN
The present paper is mainly aimed at introducing a novel notion of stability of nonlinear time-delay systems called Rational Stability. According to the Lyapunov-type, various sufficient conditions for rational stability are reached. Under delay dependent conditions, we suggest a nonlinear time-delay observer to estimate the system states, a state feedback controller and the observer-based controller rational stability is provided. Moreover, global rational stability using output feedback is given. Finally, the study presents simulation findings to show the feasibility of the suggested strategy.
EN
In the paper the parametric optimization problem for a linear system with two delays and a PD-controller is presented. In the parametric optimization problem the quadratic performance index is considered. The value of the quadratic index of quality is calculated due to the Lyapunov functional and is equal to the value of that functional for the initial function of the neutral system with two delays. The Lyapunov functional is determined by means of the Lyapunov matrix.
EN
In the paper construction of a Lyapunov functional for time delay system with both lumped and distributed delay is presented. The Lyapunov functional is determined by means of the Lyapunov matrix. The method of determination of the Lyapunov matrix for time delay system with both lumped and distributed delay is presented. It is given the example illustrating the method.
EN
In the paper a Lyapunov matrices approach to the parametric optimization problem of time-delay systems with two commensurate delays and a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of the Lyapunov functional for the initial function of time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.
EN
In the paper a Lyapunov matrices approach to the parametric optimization problem of a neutral system with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the neutral system. The Lyapunov functional is determined by means of the Lyapunov matrix.
EN
In the paper a Lyapunov matrices approach to the parametric optimization problem of timedelay systems with a P-controller is presented. The value of integral quadratic performance index of quality is equal to the value of Lyapunov functional for the initial function of the time-delay system. The Lyapunov functional is determined by means of the Lyapunov matrix.
EN
The paper presents a method of determining of the Lyapunov functional for a linear neutral system with an interval time-varying delay. The Lyapunov functional is constructed for the system with a time-varying delay with a given time derivative, which is calculated on the trajectory of the system with a time-varying delay. The presented method gives analytical formulas for the coefficients of the Lyapunov functional.
EN
In this paper a parametric optimization problem for a linear neutral system with two delays with an integral quadratic performance index is formulated and solved. The method of computing of a performance index value bases on determining of a Lyapunov functional defined on a state space such that its value for an initial state is equal to a performance index value. In the paper a form of a Lyapunov functional is assumed and a method of computing its coefficients is given. An example illustrating the application of discussed theory is presented. It concerns the system with a PD-controller designed to control a plant with two delays both retarded and neutral type. For such system a value of considered performance index is determined.
EN
The paper presents a method to determine a Lyapunov functional for a linear time-invariant system with an interval time-varying delay. The functional is constructed for the system with a time-varying delay with a given time derivative, which is calculated on the system trajectory. The presented method gives analytical formulas for the coefficients of the Lyapunov functional.
11
Content available remote Global solution of reaction diffusion system with non diagonal matrix
EN
The purpose of this paper is to prove the global existence in time of solutions for the coupled reaction-diffusion system: (…) with triangular matrix of diffusion coefficients. By combining the Lyapunov functional method with the regularizing effect, we show that global solutions exist. Our investigation applied for a wide class of the nonlinear terms f and g.
12
Content available Funkcjonały Lapunowa dla układów z opóźnieniem
PL
W pracy przedstawiono zastosowanie funkcjonałów Lapunowa do badania stabilności układów z opóźnieniem oraz w procesie optymalizacji parametrycznej.
EN
In the paper was presented the application of the Lyapunov functionals to examination of the stability of the time delay systems and in the process of parametric optimization.
EN
The paper presents a method of determining the Lyapunov quadratic functional for linear tirne-invariant system with both lumped and distributed delay. The Lyapunov functional is constructed for its given time derivative, which is calculated on the trajectory of the system with both lumped and distributed delay. The method presented gives analytical formulas for the coefficients of the Lyapunov functional.
EN
In the paper the problem of parametric optimization of a linear neutral system with two delays accomplished with respect to the integral quadratic performance index is formulated and solved. A method of computing the value of performance index relies on determining such a Lyapunov functional defined on the state space that its value for the initial state is equal to that of performance index. In the paper the form of Lyapunov functional is assumed and a method of computing its coefficients is given. An example illustrating the application of the theory discussed is presented. It concerns a system with the P-controller designed to control a plant with two delays both retarded and neutral type. For such a system the value of the considered performance index is determined.
15
Content available remote Lyapunov functional for a system with k-non-commensurate neutral time delays
EN
The paper presents a method of determining the Lyapunov quadratic functional for linear time-invariant system with k-non-commensurate neutral type time delays. The Lyapunov functional is constructed for its given time derivative, which is calculated on the trajectory of the system with k-non-commensurate neutral type time delays. The presented method gives analytical formulas for the coefficients of the Lyapunov functional.
16
Content available remote Lyapunov functional for a linear system with two delays
EN
The paper presents a method of determining the Lyapunov functional for linear time-invariant LTI system with two lumped delays.
PL
W pracy rozważono zagadnienie stabilizacji pewnej klasy nieliniowych układów o jednym wejściu i jednym wyjściu za pomocą dynamicznego sprzężenia zwrotnego. Cechą charakterystyczną rozważanych układów jest ich asymptotyczna stabilność już przy braku sterowania. W pracy pokazano, że zastosowanie dynamicznego sprzężenia zwrotnego pozwala wprowadzić do układu dodatkowe tłumienie i polepszyć własności dynamiczne układu. Do najważniejszych zalet zastosowanego schematu stabilizacji należy zaliczyć niski rząd dynamicznego sprzężenia zwrotnego oraz jego odporność na zmiany parametrów. Własność asymptotycznej stabilności układu została pokazana z wykorzystaniem odpowiednich funkcjonałów Lapunowa oraz twierdzenia LaSalle'a. Wyniki teoretyczne zostały zweryfikowane poprzez obliczenia numeryczne i symulacje komputerowe.
EN
The paper considers a class of single-input single-output (SISO) nonlinear systems whose behavior is defined by nonlinear differential equations. An uncontrolled system from this class is already asymptotically stable. Dynamie feedback controls are given to improve the dynamie stability perfonnance of appropriate closed-loop systems. The main advantages of the presented approach are reduced-order design of the controllers and their robustness. The asymptotic stability in the Lyapunov sense is analyzed and proved by the use of Lyapunov funetionals and LaSalle's invariance principle. The results of computer simulation are included to verify theoretical analysis and mathematical formulation.
18
EN
The paper addresses the problem of designing a robust output/state model predictive control for linear polytopic systems without constraints. The new robust BMI stability condition for given predictive and control horizon is derived which guarantees the parameter dependent quadratic stability and guaranteed cost.The proposed condition is appropriate for centralized and decentralized control design, as illustrated on example.
EN
The paper presents a method of determining of the Lyapunov quadratic functional for linear time-invariant system with two delays both retarded and neutral type. The Lyapunov functional is constructed for a given its time derivative which is calculated on the trajectory of the system with two delays both retarded and neutral type. The presented method gives analytical formulas for the coefficients of the Lyapunov functional.
EN
A Lur'e feedback of indirect control system consisting of a nonlinear static periodic differentiable function and a linear finite-dimensional part arising in the analysis of phased-locked loop systems in considered. We simplify and complete the results of [3, Theorem 1.2.1, p. 285] and then apply them to approximate the pull-in range of two analog phase-locked loop systems.
PL
Rozważa się układ Lurie sterowania pośredniego, złożony z różniczkowalnej, okresowej nieliniowości statycznej i skończenie wymiarowej części liniowej, powstający przy analizie pętli fazowej. Upraszcza się i uzupełnia rezultaty [3, Twierdzenie 4.2.1, str. 285], a następnie uzyskane wyniki stosuje się do aproksymacji zakresu chwytania dwóch przykładowych, analogowych pętli synchronizacji fazy.
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