Dissipative dynamical systems part I: General theory
Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Willems, Jan C. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
1972 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer-Verlag 1972 |
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| Übergeordnetes Werk: |
Enthalten in: Archive for rational mechanics and analysis - Springer-Verlag, 1957, 45(1972), 5 vom: Jan., Seite 321-351 |
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| Übergeordnetes Werk: |
volume:45 ; year:1972 ; number:5 ; month:01 ; pages:321-351 |
| Links: |
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| DOI / URN: |
10.1007/BF00276493 |
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| Katalog-ID: |
OLC2056394693 |
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| 520 | |a Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. | ||
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10.1007/BF00276493 doi (DE-627)OLC2056394693 (DE-He213)BF00276493-p DE-627 ger DE-627 rakwb eng 530 510 VZ Willems, Jan C. verfasserin aut Dissipative dynamical systems part I: General theory 1972 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1972 Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. State Space Lyapunov Function Stable Equilibrium State Space Model Dissipative System Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 45(1972), 5 vom: Jan., Seite 321-351 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:45 year:1972 number:5 month:01 pages:321-351 https://doi.org/10.1007/BF00276493 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_59 GBV_ILN_65 GBV_ILN_70 GBV_ILN_170 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2027 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 AR 45 1972 5 01 321-351 |
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10.1007/BF00276493 doi (DE-627)OLC2056394693 (DE-He213)BF00276493-p DE-627 ger DE-627 rakwb eng 530 510 VZ Willems, Jan C. verfasserin aut Dissipative dynamical systems part I: General theory 1972 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1972 Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. State Space Lyapunov Function Stable Equilibrium State Space Model Dissipative System Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 45(1972), 5 vom: Jan., Seite 321-351 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:45 year:1972 number:5 month:01 pages:321-351 https://doi.org/10.1007/BF00276493 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_59 GBV_ILN_65 GBV_ILN_70 GBV_ILN_170 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2027 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 AR 45 1972 5 01 321-351 |
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10.1007/BF00276493 doi (DE-627)OLC2056394693 (DE-He213)BF00276493-p DE-627 ger DE-627 rakwb eng 530 510 VZ Willems, Jan C. verfasserin aut Dissipative dynamical systems part I: General theory 1972 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1972 Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. State Space Lyapunov Function Stable Equilibrium State Space Model Dissipative System Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 45(1972), 5 vom: Jan., Seite 321-351 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:45 year:1972 number:5 month:01 pages:321-351 https://doi.org/10.1007/BF00276493 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_59 GBV_ILN_65 GBV_ILN_70 GBV_ILN_170 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2027 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 AR 45 1972 5 01 321-351 |
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10.1007/BF00276493 doi (DE-627)OLC2056394693 (DE-He213)BF00276493-p DE-627 ger DE-627 rakwb eng 530 510 VZ Willems, Jan C. verfasserin aut Dissipative dynamical systems part I: General theory 1972 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1972 Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. State Space Lyapunov Function Stable Equilibrium State Space Model Dissipative System Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 45(1972), 5 vom: Jan., Seite 321-351 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:45 year:1972 number:5 month:01 pages:321-351 https://doi.org/10.1007/BF00276493 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_59 GBV_ILN_65 GBV_ILN_70 GBV_ILN_170 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2027 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 AR 45 1972 5 01 321-351 |
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10.1007/BF00276493 doi (DE-627)OLC2056394693 (DE-He213)BF00276493-p DE-627 ger DE-627 rakwb eng 530 510 VZ Willems, Jan C. verfasserin aut Dissipative dynamical systems part I: General theory 1972 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer-Verlag 1972 Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. State Space Lyapunov Function Stable Equilibrium State Space Model Dissipative System Enthalten in Archive for rational mechanics and analysis Springer-Verlag, 1957 45(1972), 5 vom: Jan., Seite 321-351 (DE-627)129519618 (DE-600)212130-X (DE-576)014933004 0003-9527 nnns volume:45 year:1972 number:5 month:01 pages:321-351 https://doi.org/10.1007/BF00276493 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-TEC SSG-OLC-PHY SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_21 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_30 GBV_ILN_31 GBV_ILN_32 GBV_ILN_40 GBV_ILN_59 GBV_ILN_65 GBV_ILN_70 GBV_ILN_170 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2014 GBV_ILN_2016 GBV_ILN_2018 GBV_ILN_2020 GBV_ILN_2027 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4035 GBV_ILN_4036 GBV_ILN_4046 GBV_ILN_4082 GBV_ILN_4126 GBV_ILN_4305 GBV_ILN_4307 GBV_ILN_4310 GBV_ILN_4311 GBV_ILN_4313 GBV_ILN_4314 GBV_ILN_4315 GBV_ILN_4316 GBV_ILN_4318 GBV_ILN_4319 GBV_ILN_4323 GBV_ILN_4325 GBV_ILN_4700 AR 45 1972 5 01 321-351 |
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Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. © Springer-Verlag 1972 |
| abstractGer |
Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. © Springer-Verlag 1972 |
| abstract_unstemmed |
Abstract The first part of this two-part paper presents a general theory of dissipative dynamical systems. The mathematical model used is a state space model and dissipativeness is defined in terms of an inequality involving the storage function and the supply function. It is shown that the storage function satisfies an a priori inequality: it is bounded from below by the available storage and from above by the required supply. The available storage is the amount of internal storage which may be recovered from the system and the required supply is the amount of supply which has to be delivered to the system in order to transfer it from the state of minimum storage to a given state. These functions are themselves possible storage functions, i.e., they satisfy the dissipation inequality. Moreover, since the class of possible storage functions forms a convex set, there is thus a continuum of possible storage functions ranging from its lower bound, the available storage, to its upper bound, the required supply. The paper then considers interconnected systems. It is shown that dissipative systems which are interconnected via a neutral interconnection constraint define a new dissipative dynamical system and that the sum of the storage functions of the individual subsystems is a storage function for the interconnected system. The stability of dissipative systems is then investigated and it is shown that a point in the state space where the storage function attains a local minimum defines a stable equilibrium and that the storage function is a Lyapunov function for this equilibrium. These results are then applied to several examples. These concepts and results will be applied to linear dynamical systems with quadratic supply rates in the second part of this paper. © Springer-Verlag 1972 |
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