Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Volterra series: Identification problems and nonlinear orderseparation
Damien Bouvier1, Thomas Hélie1, David Roze1
1Project-team S3: Systems, Signals and Sound (http://s3.ircam.fr/)Science and Technology of Music and Sound
UMR 9912 Ircam-CNRS-UPMC
14 October 2016
14 October 2016 Nonlinear problems and Volterra series 1 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Context and motivations
Thesis (started in October 2015)• Goal: Realist instrumental sound synthesis (tone evolution VS. note and loudness)• Idea:
InputOutput� - Identification
Nonlinear systems under study:Distortion pedal, compressor, loudspeaker, nonlinear resonator, ...
• Fading memory• Regular nonlinearities:
‚ Taylor-like expansion‚ No chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 2 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Context and motivations
Thesis (started in October 2015)• Goal: Realist instrumental sound synthesis (tone evolution VS. note and loudness)• Idea:
InputOutput� - Identification ∆ Real-time
Synthesis
Nonlinear systems under study:Distortion pedal, compressor, loudspeaker, nonlinear resonator, ...
• Fading memory• Regular nonlinearities:
‚ Taylor-like expansion‚ No chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 2 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Context and motivations
Thesis (started in October 2015)• Goal: Realist instrumental sound synthesis (tone evolution VS. note and loudness)• Idea:
AudioDatabase
� - Identification ∆ Real-timeSynthesis
Nonlinear systems under study:Distortion pedal, compressor, loudspeaker, nonlinear resonator, ...
• Fading memory• Regular nonlinearities:
‚ Taylor-like expansion‚ No chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 2 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Context and motivations
Thesis (started in October 2015)• Goal: Realist instrumental sound synthesis (tone evolution VS. note and loudness)• Idea:
InputOutput� - Identification ∆ Real-time
Synthesis
Nonlinear systems under study:Distortion pedal, compressor, loudspeaker, nonlinear resonator, ...
• Fading memory• Regular nonlinearities:
‚ Taylor-like expansion‚ No chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 2 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Nonlinear system representation
Hammerstein
fh(.)u H xWiener
Hu fw (.) x
14 October 2016 Nonlinear problems and Volterra series 3 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Nonlinear system representation
Hammerstein
fh(.)u H xWiener
Hu fw (.) x
Hammerstein-Wiener
fh(.)u H fw (.) x
Wiener-Hammerstein
H1u f (.) H2
x
14 October 2016 Nonlinear problems and Volterra series 3 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Nonlinear system representation
Hammerstein
fh(.)u H xWiener
Hu fw (.) x
Hammerstein-Wiener
fh(.)u H fw (.) x
Wiener-Hammerstein
H1u f (.) H2
x
But
Description not adapted to physical systems
14 October 2016 Nonlinear problems and Volterra series 3 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Volterra series
Hu x
Linear system: x(t) =⁄
Rh(·) u(t ≠ ·) d·
Volterra series: x(t) =ÿ
n=1
⁄
Rnhn(·1, . . . , ·n)¸ ˚˙ ˝
Volterra kernels
u(t ≠ ·1) · · · u(t ≠ ·n) d·1 · · · d·n
Remarks• General representation for regular nonlinearities• ÷ kernel representation in the spectral domain• Interconnection laws (sum, product, cascade) in temporal & spectral domain• Representation only valid:
‚ around an equilibirum (here x0 = 0)‚ in a computable convergence domain [Hélie and Laroche, 2011]
• Cannot perform hysteresis or chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 4 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Volterra series
Hu x
Linear system: x(t) =⁄
Rh(·) u(t ≠ ·) d·
Volterra series: x(t) =ÿ
n=1
⁄
Rnhn(·1, . . . , ·n)¸ ˚˙ ˝
Volterra kernels
u(t ≠ ·1) · · · u(t ≠ ·n) d·1 · · · d·n
Remarks• General representation for regular nonlinearities• ÷ kernel representation in the spectral domain• Interconnection laws (sum, product, cascade) in temporal & spectral domain• Representation only valid:
‚ around an equilibirum (here x0 = 0)‚ in a computable convergence domain [Hélie and Laroche, 2011]
• Cannot perform hysteresis or chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 4 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Volterra series
Hu x
Linear system: x(t) =⁄
Rh(·) u(t ≠ ·) d·
Volterra series: x(t) =ÿ
n=1
⁄
Rnhn(·1, . . . , ·n)¸ ˚˙ ˝
Volterra kernels
u(t ≠ ·1) · · · u(t ≠ ·n) d·1 · · · d·n
Remarks• General representation for regular nonlinearities• ÷ kernel representation in the spectral domain• Interconnection laws (sum, product, cascade) in temporal & spectral domain• Representation only valid:
‚ around an equilibirum (here x0 = 0)‚ in a computable convergence domain [Hélie and Laroche, 2011]
• Cannot perform hysteresis or chaotic behaviour
14 October 2016 Nonlinear problems and Volterra series 4 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Summary
From linear to nonlinear systems
• Immitance inversion (impedance ¡ admittance)• Passivity• System identification
Nonlinear order separation
• Order homogeneity in Volterra• Idea 1: Using amplitude [Boyd et al., 1983]• Idea 2: Using phase• Simulation results
14 October 2016 Nonlinear problems and Volterra series 5 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Immitance inversion (impedance ¡ admittance)
Notion of immitance
Z
YFlow E�ort
Impedance
Admittance
Ccurrent (A)
velocity (m/s)acoustic flow (m3/s)
D Cvoltage (V)force (N)
acoustic pressure (Pa)
D
Linear system: Y (s) = 1Z(s)
(with s the Laplace variable)
Volterra series (using interconnection laws) [Schetzen, 1976]:
Y1 © 1Z1
and Yn © fct(Z1, . . . , Zn) for n Ø 2
Conclusion: ÷ well-defined inversion for nonlinear systems
14 October 2016 Nonlinear problems and Volterra series 6 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Immitance inversion (impedance ¡ admittance)
Notion of immitance
Z
YFlow E�ort
Impedance
Admittance
Ccurrent (A)
velocity (m/s)acoustic flow (m3/s)
D Cvoltage (V)force (N)
acoustic pressure (Pa)
D
Linear system: Y (s) = 1Z(s)
(with s the Laplace variable)
Volterra series (using interconnection laws) [Schetzen, 1976]:
Y1 © 1Z1
and Yn © fct(Z1, . . . , Zn) for n Ø 2
Conclusion: ÷ well-defined inversion for nonlinear systems
14 October 2016 Nonlinear problems and Volterra series 6 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Immitance inversion (impedance ¡ admittance)
Notion of immitance
Z
YFlow E�ort
Impedance
Admittance
Ccurrent (A)
velocity (m/s)acoustic flow (m3/s)
D Cvoltage (V)force (N)
acoustic pressure (Pa)
D
Linear system: Y (s) = 1Z(s)
(with s the Laplace variable)
Volterra series (using interconnection laws) [Schetzen, 1976]:
Y1 © 1Z1
and Yn © fct(Z1, . . . , Zn) for n Ø 2
Conclusion: ÷ well-defined inversion for nonlinear systems
14 October 2016 Nonlinear problems and Volterra series 6 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Immitance inversion (impedance ¡ admittance)
Notion of immitance
Z
YFlow E�ort
Impedance
Admittance
Ccurrent (A)
velocity (m/s)acoustic flow (m3/s)
D Cvoltage (V)force (N)
acoustic pressure (Pa)
D
Linear system: Y (s) = 1Z(s)
(with s the Laplace variable)
Volterra series (using interconnection laws) [Schetzen, 1976]:
Y1 © 1Z1
and Yn © fct(Z1, . . . , Zn) for n Ø 2
Conclusion: ÷ well-defined inversion for nonlinear systems
14 October 2016 Nonlinear problems and Volterra series 6 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Passivity (u æ Immitance æ x)
Definition (Passivity [Youla et al., 1959; Boyd and Chua, 1982])A system is passive if and only if it does not return more energy than it consumed, i.e.:
⁄ T
≠Œu(t) x(t) dt Ø 0 , ’T œ R
System Passivity criterion
u(t) æ [h ı u](t) linear Re[H(s)] Ø 0 ’s œ C+
u(t) æ u(t)n memorylesshomogeneous order Either
Ó n even ∆ not passiven odd ∆ passive
u(t) æqN
n=1 –nu(t)n memoryless Positivity of polynomial of orderN + 1 of coe�cients {pn} = {–n≠1}
u[k] æqL
l=0 hn[l]u[k ≠ l]ndiscrete time
volterra kernelhomogeneous order
Positivity of the eigenvalues ofsupersymmetric tensor Fnassociated with hn [Qi, 2005]
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 7 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Passivity (u æ Immitance æ x)
Definition (Passivity [Youla et al., 1959; Boyd and Chua, 1982])A system is passive if and only if it does not return more energy than it consumed, i.e.:
⁄ T
≠Œu(t) x(t) dt Ø 0 , ’T œ R
System Passivity criterion
u(t) æ [h ı u](t) linear Re[H(s)] Ø 0 ’s œ C+
u(t) æ u(t)n memorylesshomogeneous order Either
Ó n even ∆ not passiven odd ∆ passive
u(t) æqN
n=1 –nu(t)n memoryless Positivity of polynomial of orderN + 1 of coe�cients {pn} = {–n≠1}
u[k] æqL
l=0 hn[l]u[k ≠ l]ndiscrete time
volterra kernelhomogeneous order
Positivity of the eigenvalues ofsupersymmetric tensor Fnassociated with hn [Qi, 2005]
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 7 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Passivity (u æ Immitance æ x)
Definition (Passivity [Youla et al., 1959; Boyd and Chua, 1982])A system is passive if and only if it does not return more energy than it consumed, i.e.:
⁄ T
≠Œu(t) x(t) dt Ø 0 , ’T œ R
System Passivity criterion
u(t) æ [h ı u](t) linear Re[H(s)] Ø 0 ’s œ C+
u(t) æ u(t)n memorylesshomogeneous order Either
Ó n even ∆ not passiven odd ∆ passive
u(t) æqN
n=1 –nu(t)n memoryless Positivity of polynomial of orderN + 1 of coe�cients {pn} = {–n≠1}
u[k] æqL
l=0 hn[l]u[k ≠ l]ndiscrete time
volterra kernelhomogeneous order
Positivity of the eigenvalues ofsupersymmetric tensor Fnassociated with hn [Qi, 2005]
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 7 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Passivity (u æ Immitance æ x)
Definition (Passivity [Youla et al., 1959; Boyd and Chua, 1982])A system is passive if and only if it does not return more energy than it consumed, i.e.:
⁄ T
≠Œu(t) x(t) dt Ø 0 , ’T œ R
System Passivity criterion
u(t) æ [h ı u](t) linear Re[H(s)] Ø 0 ’s œ C+
u(t) æ u(t)n memorylesshomogeneous order Either
Ó n even ∆ not passiven odd ∆ passive
u(t) æqN
n=1 –nu(t)n memoryless Positivity of polynomial of orderN + 1 of coe�cients {pn} = {–n≠1}
u[k] æqL
l=0 hn[l]u[k ≠ l]ndiscrete time
volterra kernelhomogeneous order
Positivity of the eigenvalues ofsupersymmetric tensor Fnassociated with hn [Qi, 2005]
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 7 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Passivity (u æ Immitance æ x)
Definition (Passivity [Youla et al., 1959; Boyd and Chua, 1982])A system is passive if and only if it does not return more energy than it consumed, i.e.:
⁄ T
≠Œu(t) x(t) dt Ø 0 , ’T œ R
System Passivity criterion
u(t) æ [h ı u](t) linear Re[H(s)] Ø 0 ’s œ C+
u(t) æ u(t)n memorylesshomogeneous order Either
Ó n even ∆ not passiven odd ∆ passive
u(t) æqN
n=1 –nu(t)n memoryless Positivity of polynomial of orderN + 1 of coe�cients {pn} = {–n≠1}
u[k] æqL
l=0 hn[l]u[k ≠ l]ndiscrete time
volterra kernelhomogeneous order
Positivity of the eigenvalues ofsupersymmetric tensor Fnassociated with hn [Qi, 2005]
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 7 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
System identification
Idea: Computing the system function from input and output.
Linear systems
• Transfer function: H(s) = X(s)U(s)
• Several methods of identification, in order to improve robustness(Impulse response method, Spectral analysis, Cross-correlation method)
Nonlinear systemsB Notion of transfer function not valid
• For Volterra series: theoretical work by Boyd et al. [1983, 1984]Order 1 & 2 in practice, robustness problems
• For Hammerstein system: Farina [2000]; Rébillat et al. [2011]; Novák et al. [2010]Method robust, e�cient and quick
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 8 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
System identification
Idea: Computing the system function from input and output.
Linear systems
• Transfer function: H(s) = X(s)U(s)
• Several methods of identification, in order to improve robustness(Impulse response method, Spectral analysis, Cross-correlation method)
Nonlinear systemsB Notion of transfer function not valid
• For Volterra series: theoretical work by Boyd et al. [1983, 1984]Order 1 & 2 in practice, robustness problems
• For Hammerstein system: Farina [2000]; Rébillat et al. [2011]; Novák et al. [2010]Method robust, e�cient and quick
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 8 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
System identification
Idea: Computing the system function from input and output.
Linear systems
• Transfer function: H(s) = X(s)U(s)
• Several methods of identification, in order to improve robustness(Impulse response method, Spectral analysis, Cross-correlation method)
Nonlinear systemsB Notion of transfer function not valid
• For Volterra series: theoretical work by Boyd et al. [1983, 1984]Order 1 & 2 in practice, robustness problems
• For Hammerstein system: Farina [2000]; Rébillat et al. [2011]; Novák et al. [2010]Method robust, e�cient and quick
In general: Open problem, no solution yet (work in progress)
14 October 2016 Nonlinear problems and Volterra series 8 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
From linear to nonlinear systems
• Immitance inversion (impedance ¡ admittance)• Passivity• System identification
Nonlinear order separation
• Order homogeneity in Volterra• Idea 1: Using amplitude [Boyd et al., 1983]• Idea 2: Using phase• Simulation results
14 October 2016 Nonlinear problems and Volterra series 9 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Order homogeneity in Volterra
x(t) =ÿ
n=1
⁄
Rnhn(·1, . . . , ·n) u(t ≠ ·1) · · · u(t ≠ ·n) d·1 · · · d·n
¸ ˚˙ ˝xn(t)
Idea: Having access to the xn(t) would simplify the identification.
Multilinearity of Volterra kernels
Vnu(t) xn(t)
Vn–u(t) –nxn(t)
14 October 2016 Nonlinear problems and Volterra series 10 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Order homogeneity in Volterra
x(t) =ÿ
n=1
⁄
Rnhn(·1, . . . , ·n) u(t ≠ ·1) · · · u(t ≠ ·n) d·1 · · · d·n
¸ ˚˙ ˝xn(t)
Idea: Having access to the xn(t) would simplify the identification.
Multilinearity of Volterra kernels
Vnu(t) xn(t)
Vn–u(t) –nxn(t)
14 October 2016 Nonlinear problems and Volterra series 10 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 1: Using amplitude [Boyd et al., 1983]
Collection of input uk(t) = –ku(t).∆ the corresponding outputs are:
Âk(t) =ÿ
n
–nkxn(t)
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
–1 –21 . . . –N
1–2 –2
2 . . . –N2
......
. . ....
–N –2N . . . –N
N
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A X(t)
Advantages and disadvantages4 easy to implement6 amplitude spanning a large range measurement error
6 matrix ill-conditioned numerical error
14 October 2016 Nonlinear problems and Volterra series 11 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 1: Using amplitude [Boyd et al., 1983]
Collection of input uk(t) = –ku(t).∆ the corresponding outputs are:
Âk(t) =ÿ
n
–nkxn(t)
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
–1 –21 . . . –N
1–2 –2
2 . . . –N2
......
. . ....
–N –2N . . . –N
N
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A X(t)
Advantages and disadvantages4 easy to implement6 amplitude spanning a large range measurement error
6 matrix ill-conditioned numerical error
14 October 2016 Nonlinear problems and Volterra series 11 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 1: Using amplitude [Boyd et al., 1983]
Collection of input uk(t) = –ku(t).∆ the corresponding outputs are:
Âk(t) =ÿ
n
–nkxn(t)
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
–1 –21 . . . –N
1–2 –2
2 . . . –N2
......
. . ....
–N –2N . . . –N
N
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A X(t)
Advantages and disadvantages4 easy to implement6 amplitude spanning a large range measurement error
6 matrix ill-conditioned numerical error
14 October 2016 Nonlinear problems and Volterra series 11 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 2: Using phase
Hypothesis: Use of complex signals: u(t) œ C
Same method, with –k œ C
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
–1 –21 . . . –N
1–2 –2
2 . . . –N2
......
. . ....
–N –2N . . . –N
N
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A Y (t)
14 October 2016 Nonlinear problems and Volterra series 12 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 2: Using phase
Hypothesis: Use of complex signals: u(t) œ C
Same method, with –k œ C
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
–1 –21 . . . –N
1–2 –2
2 . . . –N2
......
. . ....
–N –2N . . . –N
N
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A Y (t)
14 October 2016 Nonlinear problems and Volterra series 12 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 2: Using phase
Hypothesis: Use of complex signals: u(t) œ C
Same method, with –k œ CSpecial case: – unit root: –k = e2ifi(k≠1)/N = wk≠1
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
1 1 . . . 1w w2 . . . 1...
.... . .
...wN w2N . . . 1
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A¸˚˙˝DFT matrix
Y (t)
14 October 2016 Nonlinear problems and Volterra series 12 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Idea 2: Using phase
Hypothesis: Use of complex signals: u(t) œ C
Same method, with –k œ CSpecial case: – unit root: –k = e2ifi(k≠1)/N = wk≠1
∆
S
WWU
Â1Â2...
ÂN
T
XXV(t) =
S
WWU
1 1 . . . 1w w2 . . . 1...
.... . .
...wN w2N . . . 1
T
XXV
S
WWU
x1x2...
xN
T
XXV(t)
�(t) = A¸˚˙˝DFT matrix
Y (t)
Advantages and disadvantages4 only one amplitude less measurement error
4 DFT matrix is well-conditioned4 Can use FFT numerical computation
6 Need for complex signals only theoretical
14 October 2016 Nonlinear problems and Volterra series 12 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Simulation results (1)
14 October 2016 Nonlinear problems and Volterra series 13 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Simulation results (2)
14 October 2016 Nonlinear problems and Volterra series 14 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Simulation results (2)
14 October 2016 Nonlinear problems and Volterra series 15 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Conclusion
Volterra series, a useful and well-known system representation ...• General paradigm for weakly nonlinear system• Theory well developed: Volterra; Brockett; Schetzen; Rugh; Boyd• Permits real-time computation
... with still a lot of open questions• Still no criterion for system passivity (work in progress)• No general and robust method for identification• Nonlinear order separation is still not resolved:
adaptation of the "phase" method for real input (work in progress)
14 October 2016 Nonlinear problems and Volterra series 16 / 16
Presentation and context From linear to nonlinear systems Nonlinear order separation Conclusion
Conclusion
Volterra series, a useful and well-known system representation ...• General paradigm for weakly nonlinear system• Theory well developed: Volterra; Brockett; Schetzen; Rugh; Boyd• Permits real-time computation
... with still a lot of open questions• Still no criterion for system passivity (work in progress)• No general and robust method for identification• Nonlinear order separation is still not resolved:
adaptation of the "phase" method for real input (work in progress)
14 October 2016 Nonlinear problems and Volterra series 16 / 16
Bibliographie References
Bibliographie I
Thomas Hélie and Béatrice Laroche. Computation of convergence bounds for Volterraseries of linear-analytic single-input systems. Automatic Control, IEEE Transactionson, 56(9):2062–2072, 2011.
Stephen Boyd, YS Tang, and Leon O Chua. Measuring volterra kernels. Circuits andSystems, IEEE Transactions on, 30(8):571–577, 1983.
Martin Schetzen. Theory of pth-order inverses of nonlinear systems. Circuits andSystems, IEEE Transactions on, 23(5):285–291, 1976.
Dante C Youla, LJ Castriota, and Herbert J Carlin. Bounded real scattering matricesand the foundations of linear passive network theory. Circuit Theory, IRETransactions on, 6(1):102–124, 1959.
S Boyd and Leon O Chua. On the passivity criterion for lti N-ports. InternationalJournal of Circuit Theory and Applications, 10(4):323–333, 1982.
Liqun Qi. Eigenvalues of a real supersymmetric tensor. Journal of SymbolicComputation, 40(6):1302–1324, 2005.
Stephen Boyd, Leon O Chua, and Charles A Desoer. Analytical foundations of Volterraseries. IMA Journal of Mathematical Control and Information, 1(3):243–282, 1984.
Angelo Farina. Simultaneous measurement of impulse response and distortion with aswept-sine technique. In Audio Engineering Society Convention 108. AudioEngineering Society, 2000.
14 October 2016 Nonlinear problems and Volterra series 17 / 16
Bibliographie References
Bibliographie II
Marc Rébillat, Romain Hennequin, Etienne Corteel, and Brian FG Katz. Identificationof cascade of Hammerstein models for the description of nonlinearities in vibratingdevices. Journal of Sound and Vibration, 330(5):1018–1038, 2011.
Antonín Novák, Laurent Simon, Frantiöek Kadlec, and Pierrick Lotton. Nonlinearsystem identification using exponential swept-sine signal. IEEE Transactions onInstrumentation and Measurement, 59(8):2220–2229, 2010.
Vito Volterra. Theory of functionals and of integral and integro-di�erential equations.Courier Corporation, 2005.
Roger W Brockett. Volterra series and geometric control theory. Automatica, 12(2):167–176, 1976.
Martin Schetzen. The Volterra and Wiener theories of nonlinear systems. {John Wiley& Sons}, 1980.
Wilson John Rugh. Nonlinear system theory. Johns Hopkins University PressBaltimore, 1981.
Stephen Poythress Boyd. Volterra series: Engineering fundamentals. PhD thesis,University of California, Berkeley, 1985.
14 October 2016 Nonlinear problems and Volterra series 17 / 16
