In this monograph are presented results of the author’s research on the determination of the Lyapunov functionals for linear systems with time delay and its applications in the parametric optimization problem. The Lyapunov quadratic functionals are used to calculation of a value of a quadratic performance index of quality in the process of parametric optimization for time delay systems. The value of that functional at the initial state of the time delay system is equal to the value of a quadratic performance index of quality. To calculate the value of a performance index of quality one needs the formulas for the Lyapunov functional coefficients. In this monograph the method proposed by J. Repin is applied to obtain the Lyapunov functionals, with coefficients given by analytical formulas. In chapter 2 are considered systems with the retarded type time delay. This method is applied to the system with one delay (chapter 2.2), to the system with two delays (chapter 2.3), to the retarded type time delay system with both lumped and distributed delay (chapter 2.4), to the system with a retarded type time-varying delay (chapter 2.5). In chapter 3 are considered neutral systems. Repin’s method is applied to the neutral system with lumped delay (chapter 3.2), to the neutral system with both lumped and distributed delay (chapter 3.3) and to the neutral system with a time-varying delay (chapter 3.4). In last years a method of determination of a Lyapunov functional by means of Lyapunov matrices is very popular. This method is applied to the parametric optimization problem of retarded type time delay system both with one and two delays (chapter 4) and to the parametric optimization problem of neutral type time delay system for system with one and two delays (chapter 5). The examples of using of the Lyapunov functionals to calculation of the performance index value in the parametric optimization problem for linear systems with time delay are given.

W książce przedstawiono zastosowania kwadratowych funkcjonałów Lapunowa do badania stabilności liniowych układów z opóźnieniem oraz w procesie optymalizacji parametrycznej układów regulacji z obiektem liniowym z opóźnieniem i regulatorami PID. Wartość wskaźnika jakości, postaci całka z kwadratu błędu, wyznaczana jest z pewnej własności kwadratowego funkcjonału Lapunowa. Mianowicie wartość funkcjonału dla stanu początkowego układu z opóźnieniem równa jest wartości wskaźnika jakości. Z kolei obszar dodatniej określoności tego funkcjonału, w przestrzeni wyznaczonej przez nastawy regulatora PID, jest obszarem asymptotycznej stabilności układu regulacji z obiektem mającym opóźnienie. W książce zaprezentowano dwie metody wyznaczenia kwadratowego funkcjonału Lapunowa. Pierwsza metoda, która zaproponowana została przez J. Repina, polega na tym, że zostaje założona postać funkcjonału kwadratowego, następnie liczona jest jego pochodna względem czasu wzdłuż trajektorii systemu z opóźnieniem i przyrównywana jest do ujemnie określonej formy kwadratowej. W ten sposób otrzymuje się układ równań algebraiczno różniczkowych, z których są wyznaczane współczynniki funkcjonału. Druga metoda, używana m.in. przez W. Charitonowa, polega na tym, że wyliczana jest całka z kwadratu rozwiązania równania różniczkowego z opóźnieniem wyrażonego przez macierz fundamentalnych rozwiązań dla przedziału czasu od zera do nieskończoności. Całka ta jest równa funkcjonałowi kwadratowemu i wyrażona jest przez odpowiednio zdefiniowaną macierz Lapunowa. Macierz Lapunowa wyznaczana jest z układu równań różniczkowych i algebraicznych.

Opisane metody wyznaczania funkcjonału kwadratowego zostały zaprezentowane dla następujących przypadków:

układ z jednym opóźnieniem skupionym,

układ z dwoma opóźnieniami skupionymi,

układ z opóźnieniem rozłożonym,

układ z opóźnieniem zmiennym w czasie,

układ neutralny z jednym opóźnieniem skupionym,

układ neutralny z opóźnieniem rozłożonym,

układ neutralny z opóźnieniem zmiennym w czasie,

układ neutralny z dwoma opóźnieniami skupionymi.

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