On the link between storage functions of allpass systems and - PowerPoint PPT Presentation

On the link between storage functions of allpass systems and Gramians Chayan Bhawal, Debasattam Pal and Madhu N. Belur Control and Computing Group (CC Group) Department of Electrical Engineering Indian Institute of Technology Bombay IEEE Conference on Decision and Control, Melbourne December 14, 2017 C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 1 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Dissipative systems and storage function A dissipative system 1 has no source of 2 can only absorb 3 can store energy. energy. energy. Rate-change-stored-energy + Dissipated power = Power supplied � �� � � �� � � �� � d Q ∆ ( w ) Q Σ ( w ) dt Q Ψ ( w ) d Dissipative systems : dt Q Ψ ( w ) � Q Σ ( w ). Power supplied with respect to system variable: Q Σ ( w )= w T Σ w . � 0 � I w = 2 u T y . e.g. Supply rate: w = ( u, y ) : Q Σ ( w ) = w T I 0 Stored energy: Storage function 1 Q Ψ ( w ) = x T Kx . 1Trentelman and Willems, SCL, Every storage function is a state function, 1997. C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 2 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Dissipative systems and storage function A dissipative system 1 has no source of 2 can only absorb 3 can store energy. energy. energy. Rate-change-stored-energy + Dissipated power = Power supplied � �� � � �� � � �� � d Q ∆ ( w ) Q Σ ( w ) dt Q Ψ ( w ) d Dissipative systems : dt Q Ψ ( w ) � Q Σ ( w ). Power supplied with respect to system variable: Q Σ ( w )= w T Σ w . � 0 � I w = 2 u T y . e.g. Supply rate: w = ( u, y ) : Q Σ ( w ) = w T I 0 Stored energy: Storage function 1 Q Ψ ( w ) = x T Kx . 1Trentelman and Willems, SCL, Every storage function is a state function, 1997. C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 2 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Dissipative systems and storage function A dissipative system 1 has no source of 2 can only absorb 3 can store energy. energy. energy. Rate-change-stored-energy + Dissipated power = Power supplied � �� � � �� � � �� � d Q ∆ ( w ) Q Σ ( w ) dt Q Ψ ( w ) d Dissipative systems : dt Q Ψ ( w ) � Q Σ ( w ). Power supplied with respect to system variable: Q Σ ( w )= w T Σ w . � 0 � I w = 2 u T y . e.g. Supply rate: w = ( u, y ) : Q Σ ( w ) = w T I 0 Stored energy: Storage function 1 Q Ψ ( w ) = x T Kx . 1Trentelman and Willems, SCL, Every storage function is a state function, 1997. C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 2 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Dissipative systems and storage function A dissipative system 1 has no source of 2 can only absorb 3 can store energy. energy. energy. Rate-change-stored-energy + Dissipated power = Power supplied � �� � � �� � � �� � d Q ∆ ( w ) Q Σ ( w ) dt Q Ψ ( w ) d Dissipative systems : dt Q Ψ ( w ) � Q Σ ( w ). Power supplied with respect to system variable: Q Σ ( w )= w T Σ w . � 0 � I w = 2 u T y . e.g. Supply rate: w = ( u, y ) : Q Σ ( w ) = w T I 0 Stored energy: Storage function 1 Q Ψ ( w ) = x T Kx . 1Trentelman and Willems, SCL, Every storage function is a state function, 1997. C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 2 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Dissipative systems and storage function A dissipative system 1 has no source of 2 can only absorb 3 can store energy. energy. energy. Rate-change-stored-energy + Dissipated power = Power supplied � �� � � �� � � �� � d Q ∆ ( w ) Q Σ ( w ) dt Q Ψ ( w ) d Dissipative systems : dt Q Ψ ( w ) � Q Σ ( w ). Power supplied with respect to system variable: Q Σ ( w )= w T Σ w . � 0 � I w = 2 u T y . e.g. Supply rate: w = ( u, y ) : Q Σ ( w ) = w T I 0 Stored energy: Storage function 1 Q Ψ ( w ) = x T Kx . 1Trentelman and Willems, SCL, Every storage function is a state function, 1997. C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 2 / 18

Storage functions of allpass/lossless systems Dissipative systems and storage function Passivity and bounded-real systems Dissipative systems d � x T Kx � � Q Σ ( w ) , where w = ( u, y ) . dt Positive-real Bounded-real � 0 � � I � 0 I Supply rate Σ I 0 0 − I Dissipation � � � � x T Kx � 2 u T y x T Kx � u T u − y T y d d inequality dt dt Conservative � � � � x T Kx = 2 u T y x T Kx = u T u − y T y d d dt dt counterpart Name Lossless Allpass s G ( s ) = s − 1 G ( s ) = s 2 +1 s +1 Example (LC circuits) (allows all frequencies to pass) C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 3 / 18

Storage functions of allpass/lossless systems Conservative systems Passivity and bounded-real systems Conservative systems d � x T Kx � = Q Σ ( w ) , where w = ( u, y ) . dt Positive-real Bounded-real � 0 � � I � 0 I Supply rate Σ I 0 0 − I Dissipation � � � � x T Kx � 2 u T y x T Kx � u T u − y T y d d inequality dt dt Conservative � � � � x T Kx = 2 u T y x T Kx = u T u − y T y d d dt dt counterpart Name Lossless Allpass s G ( s ) = s − 1 G ( s ) = s 2 +1 s +1 Example (LC circuits) (allows all frequencies to pass) C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 3 / 18

Storage functions of allpass/lossless systems Conservative systems Passivity and bounded-real systems Conservative systems d � x T Kx � = Q Σ ( w ) , where w = ( u, y ) . dt Positive-real Bounded-real � 0 � � I � 0 I Supply rate Σ I 0 0 − I Dissipation � � � � x T Kx � 2 u T y x T Kx � u T u − y T y d d inequality dt dt Conservative � � � � x T Kx = 2 u T y x T Kx = u T u − y T y d d dt dt counterpart Name Lossless Allpass s G ( s ) = s − 1 G ( s ) = s 2 +1 s +1 Example (LC circuits) (allows all frequencies to pass) C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 3 / 18

Storage functions of allpass/lossless systems Allpass systems Allpass systems System B : d dt x = Ax + Bu , y = Cx + Du , (Minimal) A ∈ R n × n , C, B T ∈ R n × p and D ∈ R p × p . Necessary condition for allpass: I − D T D = 0. � x T Kx � = u T u − y T y. d Dissipation equation: dt Linear Matrix Equations (LME): Allpass if and only if there exists K = K T ∈ R n × n � A T K + KA + C T C � KB + C T = 0 B T K + C 0 Rewriting the LME: Allpass if and only if there exists K = K T ∈ R n × n A T K + KA + C T C = 0 KB + C T = 0 C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 4 / 18

Storage functions of allpass/lossless systems Allpass systems Allpass systems System B : d dt x = Ax + Bu , y = Cx + Du , (Minimal) A ∈ R n × n , C, B T ∈ R n × p and D ∈ R p × p . Necessary condition for allpass: I − D T D = 0. � x T Kx � = u T u − y T y. d Dissipation equation: dt Linear Matrix Equations (LME): Allpass if and only if there exists K = K T ∈ R n × n � A T K + KA + C T C � KB + C T = 0 B T K + C 0 Rewriting the LME: Allpass if and only if there exists K = K T ∈ R n × n A T K + KA + C T C = 0 KB + C T = 0 C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 4 / 18

Storage functions of allpass/lossless systems Allpass systems Allpass systems System B : d dt x = Ax + Bu , y = Cx + Du , (Minimal) A ∈ R n × n , C, B T ∈ R n × p and D ∈ R p × p . Necessary condition for allpass: I − D T D = 0. � x T Kx � = u T u − y T y. d Dissipation equation: dt Linear Matrix Equations (LME): Allpass if and only if there exists K = K T ∈ R n × n � A T K + KA + C T C � KB + C T = 0 B T K + C 0 Rewriting the LME: Allpass if and only if there exists K = K T ∈ R n × n A T K + KA + C T C = 0 KB + C T = 0 C.Bhawal, D.Pal, M.Belur (CC Grp.) (Control and Computing Group (CC Group)Department of Electrical EngineeringIndian Allpass systems and Gramian EE Dept. IIT Bombay 4 / 18