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Banery wysuwane

Biało-niebieska okładka z wzorami matematycznymi
The Quadratic Functionals for Linear Time Delay Systems

Autor
Kategoria produktu
Nauki matematyczne » Matematyka
ISBN
978-83-66727-85-4
e-ISBN
978-83-67427-47-0
Typ publikacji
monografia
Format
B5
Oprawa
miękka
Liczba stron
291
Rok wydania
2022
Wydanie
1
Opis

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.
Spis treści

Summary 7
Streszczenie 8
Notations and symbols 9
1. Introduction 11
2. Quadratic functionals for linear retarded type time delay system 15
2.1.1. Mathematical model of a linear time delay system 15
2.1.2. Stability concept 16
2.1.3. The Lyapunov functional 17
2.1.4. Necessary and sufficient condition of stability 18
2.1.5. Calculation of the value of the performance index 18
2.2. The quadratic functional for a linear system with one delay 19
2.2.1. Mathematical model of a linear time delay system with one delay 19
2.2.2. Determination of the quadratic functional 21
2.2.3. The examples of the parametric optimization 25
2.2.3.1. The parametric optimization of the inertial system with delay and a P-controller 25
2.2.3.2. The parametric optimization of the inertial system with delay and an I-controller 35
2.3. The quadratic functional for a linear system with two delays 45
2.3.1. Mathematical model of a linear system with two delays 45
2.3.2. Determination of the functional 46
2.3.3. Solution of the set of differential equations (2.189) for commensurate delays 50
2.3.4. The example 52
2.4. A linear system with both lumped and distributed retarded type time delay 54
2.4.1. Mathematical model of a linear system with both lumped and distributed retarded type time delay 54
2.4.2. Determination of the quadratic functional 55
2.4.3. The examples 59
2.4.3.1. The example 1 59
2.4.3.2. The example 2 64
2.5. A linear system with a retarded type time-varying delay 69
2.5.1. Mathematical model of a linear system with a retarded type time-varying delay 69
2.5.2. Determination of the quadratic functional 70
2.5.3. The examples 74
2.5.3.1. The inertial system with delay and a P-controller 74
2.5.3.2. The example. Two dimensional system 77
2.6. The more general quadratic functional for a linear system with one delay 83
2.6.1. Mathematical model of a linear time delay system with one delay 83
2.6.2. Determination of the quadratic functional 83
2.6.3. The example. The problem of parametric optimization for separately excited D.C. motor angular velocity control system 87
2.6.3.1. Optimization results 105
3. A linear neutral system 107
3.1. Preliminaries 107
3.2. A linear neutral system with lumped delay 110
3.2.1. Mathematical model of a linear neutral system with lumped delay 110
3.2.2. Determination of the quadratic functional for a neutral system with one delay 111
3.2.3. Calculation of the value of performance index 114
3.2.4. The example of the parametric optimization of neutral system 115
3.2.4.1. The parametric optimization of the inertial system with delay and a PD-controller 115
3.3. The quadratic functional for a neutral system with both lumped and distributed time delay 119
3.3.1. Mathematical model of a linear neutral system with both lumped and distributed time delay 119
3.3.2. Determination of the quadratic functional coefficients 120
3.3.3. The example 125
3.4. A linear neutral system with a time-varying delay 127
3.4.1. Mathematical model of a linear neutral system with a time-varying delay 127
3.4.2. Determination of the quadratic functional 129
3.4.3. The example. The inertial system with delay and a PD-controller 135
4. The Lyapunov matrix for a retarded type time delay system 141
4.1. Mathematical model of a retarded type time delay system 141
4.2. The quadratic functional for a retarded type time delay system 142
4.3. Calculation of the value of the performance index 145
4.4. The Lyapunov matrix for a system with one delay 146
4.5. The examples of the parametric optimization 148
4.5.1. The parametric optimization problem for the inertial system with delay and a P-controller 148
4.5.1.1. The system with zero equilibrium point 148
4.5.1.2. The system with non zero equilibrium point 151
4.5.2. The parametric optimization of a second order time delay system with a P-controller 156
4.5.3. The parametric optimization of a second order time delay system with a PD-controller 161
4.5.4. The parametric optimization problem for the inertial system with delay and a PI-controller 166
4.5.5. The parametric optimization of a second order time delay system with a PI-controller 174
4.5.6. The parametric optimization of a second order time delay system with a PID-controller 185
4.6. The Lyapunov matrix for a system with two commensurate delays 193
4.7. The examples of the parametric optimization 196
4.7.1. The parametric optimization problem for a scalar system with two delays and a P-controller 196
4.7.2. The parametric optimization problem for a scalar system with two delays and a PI-controller 201
4.8. The Lyapunov matrix for a system with both lumped and distributed delay 210
4.8.1. Mathematical model of a time delay system with both lumped and distributed delay 210
4.8.2. The quadratic functional 211
4.8.3. A Lyapunov matrix for system (4.250) 212
4.8.4. The example 215
5. The Lyapunov matrix for a neutral system 217
5.1. The neutral system with one delay 217
5.1.1. Mathematical model of a neutral system with one delay 217
5.1.2. The quadratic functional for a neutral system with one delay 219
5.1.3. Calculation of the value of the performance index 222
5.1.4. The Lyapunov matrix for a neutral system with one delay 223
5.1.5. The examples of the parametric optimization 225
5.1.5.1. The parametric optimization problem for a linear neutral system with a P-controller 225
5.1.5.2. The parametric optimization problem for the inertial system with delay and a PD-controller 228
5.1.5.3. The parametric optimization problem for the inertial system with delay and a PID-controller 234
5.1.5.4. The parametric optimization problem for the second order time delay system with a PD-controller 240
5.2. The neutral system with two delays 247
5.2.1. Mathematical model of neutral system with two delays 247
5.2.2. The quadratic functional for a neutral system with two delays 250
5.2.3. The Lyapunov matrix for a neutral system with two delays 252
5.2.4. The Lyapunov matrix for a neutral system with two commensurate delays 255
5.2.5. The parametric optimization problem for a system with two delays 259
5.2.6. The examples of the parametric optimization 262
5.2.6.1. The parametric optimization problem for a neutral system with two delays and a P-controller 262
5.2.6.2. The parametric optimization problem for a neutral system with two delays and a PD-controller 266
6. Conclusion 273
7. Appendix 277
A dynamical system 277
Nonlinear time delay system 277
A C0-semigroup 278
Linear time delay system 279
List of Figures 281
List of Tables 285
Bibliography 287

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